We have a specific example, in fact.
Doji Satsume was Crane Clan Champion, Emerald Champion and Doji Daimyo.
Toshimoko and Yoshi talked (pestered) him into resigning the former post in favour of his daughter Hotaru, but he held all three for a non-trivial period of time.
The Imperial Advisor doesn't really have specific servants. They (I could say 'she' as both the ones we're aware of are female) exist to provide the Emperor with a private confidante they can discuss matters with without the rules and protocol of formal court.
They have a huge amount of influence due to daily access to the Emperor, but we haven't really seen any sign they have a specific role or much legal power outside this in the way that, say, the Chancellor (who chairs the court and decides what petitions make it onto the agenda) does.

The advisor title doesn't specifically give you power - but as you are 'advisor to [insert-name-here]-sama' you get some of their authority by proxy. I believe an advisor can appoint a Yojimbo bodyguard, too.
If you're still a daimyo, you have authority over whatever town or castle or vassal family you are daimyo of.

You don't automatically lose any other title. You might give it up to have enough time to dedicate yourself to your new duties, though.

If it was the Emperor himself, then...well, he'd be the Emperor. Even the most senior and revered sensei in the dojo would treat him as such, considering it an enormous honor to have the Son of Heaven in his presence to begin with. That said, a reasonable Emperor would probably say something along the lines of, I'm here to learn from you, so do what you need to do to teach me...or he wouldn't likely have gone there in the first place. It would still be done with the utmost respect, of course, but the sensei would do whatever possible to honor the Emperor's wishes. Now, an unreasonable Emperor--like, say, Hantei XVI, the Steel Chrysanthemum--could easily make such a situation a living **** for the sensei and everyone else around. Such is life in Rokugan when the Emperor is around.
I guess a related question is, would the Emperor go to a dojo to be trained in stuff to begin with? Probably not--at least to some random dojo somewhere in the Empire--but he certainly could, if he wanted to. Because he's, you know, the Emperor...
It would be similar for the Emperor's son. However, the Emperor (dad) stands to be a moderating influence in this case, since he would likely take a dim view of his son being a jerk, because that reflects badly on the Hantei name. Whether he actually DOES or not is another matter, of course...but he certainly could. As noted, having the Emerald Champion as the son's sensei simplifies things, because it would be much more of a regular master-student relationship; the EC has that kind of status.
So...kinda fuzzy answer, because it's an unusual situation to which everyone would have to adapt the regular protocols around the Emperor.

If I understand your question, you're asking who would show greater reverence, the son of the Emperor or the sensei of the dojo? Generally, the sensei would, simply because the prince will have higher status. However, I could certainly see specific situations--for example, the sensei is a renowed war hero who performed great deeds in service to the Empire--in which the prince may choose to show greater respect than normal for the sensei. He probably wouldn't bow lower, but he might bow more deeply than strict protocol demands.
But the prince does remain the prince, regardless.

The Emporers son is trained personally by the emerald champion. In one of the novelettes he is being trained by the ruby champion (right hand samurai of the emerald) and the Emporers son has clear authority over her. So I imagine he would have authority over the emerald as well. 
 
Well I don't know if that authority extends to being able to tell them their business or how to do their jobs ect. But the hier clearly has enough authority to get them to be  polite and subservient to his wishes. He's not the emporer but is still descended from the Kami and mandated by heaven. (Unless of course the mandate of heaven has been lost :0 blasphemy! Seppuku!)

He's talking about 4th Edition, not FFG.

The sourcebooks will add more schools (and families and clans) as they go on.
To date, the only imperial schools are the Miya Herald & Cartographer, Otomo Schemer, and Seppun Astrologer & Palace Guard.
Which schools did you have in mind?
Note that a member of the Imperial Herald school is not the Imperial Herald (not sure if this is what you mean).
As and when we get stuff like Emerald Champion, Imperial Chancellor, Imperial Advisor, etc, they will most likely come into the game as titles rather than schools.

The Emperor owns everything. At least on paper and in the eyes of heaven. I believe that all the Clans pay leases/taxes every year to the Imperial Throne. And those taxes largely fund the Imperial Families. As noted the families also have various (usually minor) holding that also generate at least some revenue. And the Imperial Court can offload some of their upkeep with the Winter Court.
On a practical level I would assume that the Imperial families have Diamyo that are as rich as their Great Clan counterparts. Per EE the Miya start with 8 koku, The Otomo with 9 koku, and the Seppun start with 6 koku. That suggests that Miya and the Otomo are among the richest families (at least per capita) with the Seppun lagging somewhat behind...though not poor! The Seppun still start with more wealth than any of the families of the Crab*, Dragon, Phoenix or Lion clans. 
On a practical level, if an Imperial Daimyo wants it, they have it. 
*excluding the Yasuki

There has never been much information on it which can be both a blessing and a curse.
This is how I would currently handle it, more from a lore pespective than anything else:
Hantei: They don't seem to have any direct holding. Even Otosan Uchi is partitioned to governors, although as these seem to be exclusively appointed positions rather than hereditary we can assume there are taxes being levied on the city for them. This is extremely unusual and with no real world equivalent for non-figurehead rulers.
Maybe the City of the Rich Frog counts as one of their direct Holdings.
Speaking of taxes we also don't have much information on them. The lack of direct Hantei holdings is a bit of an anomaly if you compare it with real world examples and in feudal societies charging taxes on vassal was not done unusual complicated. There is also the Rokugani focus on rice which is even more extreme than that of Tokugawa Japan, and rather a-historical to say the least. Lets just say there is, probably, some taxing being done over the clans.
Using some tidbits from both Emerald Empire and the old pre-FFG lore I would say that some of the expenses of the court end up being indirectly outsourced to the clans.
For example the maintenance of Imperial Highways and Waystation towns. For an example in Tokugawa Japan they were mantained by the Shogunate and subsidized but I can in Rokugan having to be mantained by the clans while providing income directly to the Imperial Court. Labour corvées for Imperial Holding provided by the clans could be another example which has historical precedent in many places, including Japan. We have a direct example in the fact that the Tortoise used to charged with firefighting in Otosan Uchi.
Another example is Winter Court. Much like Royal Progresses they allow the Hantei to do some proximity rulership or "show the flag", so to speak, in different parts of the country. Since the hosting cost are the exclusive responsability of the host clan, so the Hantei manage to oth not have any upkeep for about four months every year and levy an additional economic burden on a possible rival.
In a gift-giving society like Rokugan this can also be used to work as a form of taxation while mantaining the outward appearance it isn't. Again, the Tokugawa provide a good example of this, but the Chinese tributary system can also be used as a model.
Seppun: They have their own lands on what seems to be a in fairly fertile area. If the lore hasn't changed they also hold Koutetsukan on the Islands of Silk and Spice. I assume most of their income come from these holdings.
Otomo: If there have been no changes from the old lore, the Otomo have no holdings other than their palace in the Imperial Mound, so that means they would be subsidized by the Hantei themselves.
Having said that, in the old lore after a specific event, the Otomo were given lands formerly belonging to the Crane. Personally I don't like how the Hantei and their direct vassals lack holdings despite being shown to be more than figureheads, so I would retroactively make the Otomo own that holding from the beggining and be their main source of income.
Miya: They have their own lands Southwest of the Spine of the World Mountains. It doesn't seem to be a very developed area, but it is very large and with apparent room for expansion, so I would say that the Miya are potentially the highest income earners of the Imperial families.
 
Looking back the Lands of the Imperial Champion, while intended to support the EC may also contribute for Imperial income in the periods where there is no Champion appointed.
As for incomes, therehas never been any example. For the historical examples when Daimyo got an actual legal definition during the Azuchi-Momoyama period they were landowners with lands valued at at least 10000 Koku. The domains of the Tokugawa shogunate were 5 million koku.
Whether that applies to your Rokugan or not is for you to decide.
 

Some taxes are set at an imperial level (the Emerald Champion was reviewing taxes on barley whilst waiting for the Emperor in Tiger Stalks His Prey) which presumably means they're collected by or at least for the Imperial Families. 
The Imperial Families don't really earn much directly - the Emperor basically owns the whole shooting match and lets people farm/live in/trade on his rivers, fields, cities and roads in exchange for tribute and taxes. There are a few purely 'imperial' domains directly administered by an imperial governor (like The City of the Rich Frog), where most of the city's revenue will presumably go directly into imperial coffers. 
How much any of the Daimyo has as a personal allowance/stipend/whatever from the Emperor and their own estates? Not a clue, aside from "a shed-load".
@TheWanderingJewels put up a nice exploration of what Rokugani currency actually means here: 
Which suggests a nominal amount to draw on of about 5,000 Koku per year as a personal discretionary wealth (i.e. without having to record it or justify it to anyone else in the family beyond "want that one"). If you're after something as the Miya Daimyo rather than just Miya Satoshi as an individual.....frankly the sky's the limit. 
 

It also depends on the situation. Who has more power in what situation?
The line of feudal authority runs
Heaven > Emperor > Clan Champion > Clan Family Daimyo > Clan Provincial Daimyo > Samurai from that province, so whilst a Miya or Otomo courtier may be an honoured guest in some rural castle, they're not supposed to be giving the clan samurai there orders. 
By comparison, in terms of defining imperial law (which is nominally binding on everyone in the empire) the Imperial Families are pretty much unchallengable, since even if a clan wins the debate in court and sets the policy on a new law, it'll be the scribes and scholars of the Imperial Families who actually write the wording, replete with carefully hidden loopholes. 
 
In terms of raw 'status', check page 306 of the rulebook.
Daimyo of Imperial Families are considered higher status than a Great Clan Champion, as are most posts with "Imperial" in their title. A generic Imperial Family samurai, less so (Emerald Empire starts a no-name Otomo, Miya or Seppun samurai as status 40, equivalent to most magistrates).
 
The list of Imperial posts in the status section are:
The Emperor (Currently Hantei Jodan, Hantei XXXVIII) The Imperial Spouse (Currently Hantei Hochiahime) The Voice Of The Emperor (??? in previous editions of the game this was the emperor's personal diplomat, but since the post came into being post Scorpion Coup, wasn't Imperial Family) The Emperor's Children (Currently Hantei Sotorii and Hantei Daisetsu) The Emerald Champion (Currently Akodo Toturi) The Jade Champion (vacant) The Imperial Advisor (Currently Bayushi Kachiko) The Imperial Chancellor (Currently Kakita Yoshi) The Imperial Herald (not known but traditionally Miya Family Daimyo) The Imperial Treasurer (not known)

It really relies on how each clan leverages that access and power they have.  Though the Imperial families sit above the Great Clans officially, their power is dependent on what the Emperor is going to allow them to have.  As the Imperial Families have immediate royal blood, giving them too much power can be detrimental to the Emperor as it can allow for military coups.  At the same time, they still need to be able to defend the Emperor (and themselves) against the Great Clans.  This is a bit of a fine line with some families like the Miya who have a reputation of being hostages.

By the "chosen of the emperor", I'm assuming you mean the Emerald Champion, the Imperial Advisor and the like. It actually varies. The Emerald Champion wields enormous power, being not just the Emperor's champion, but also the Empire's chief magistrate, the commander of the Imperial Legions and the head of a big chunk of the Imperial bureaucracy (not bad for a position generally selected by a sword-fight, when you think about it). So the EC is...well, far more powerful, and has much higher status, than anyone except the Emperor himself. Now, the EC generally doesn't actually DO all those jobs, and delegates a lot of it. But the authority being delegated is his/hers.
The Imperial Advisor is also extremely powerful, but in a much "softer" way. The Advisor oversees some of the bureaucracy, if memory serves (at least, they did in the old lore; I wrote that part of the old "Secrets of the Empire" book, about the Imperial bureaucracy, and I seem to remember assigning some of the bureaucracy to the Advisor...but I might be wrong and don't have the book handy to look it up). However, the Advisor has direct and immediate access to the Emperor, and can (potentially, at least; it would depend on the particular Emperor) influence the decisions that come from the Throne is pretty fundamental ways.
I'm not sure if we currently have an Imperial Treasurer; it's really a much more bureaucratic position, though, so is powerful within a pretty narrow set of boundaries dealing with things like taxation, monetary policy, etc.
We don't have a Jade Champion or Shogun currently (frankly, I'm not sure if the latter position is even recognized in the new lore...I don't recall it being mentioned anywhere, but again, I could be wrong).
The Imperial families, OTOH, are also quite powerful in their own ways. None of them really come close to the clout of the Emerald Champion, but the Otomo are firmly embedded in the bureaucracy, the Seppun are responsible for the safety and welfare of the Emperor and his family, and the Miya are the Emperor's "eyes, ears and voice" throughout the Empire. The daimyo of each of these families has access to the Emperor similar to that of the Advisor, so has (again, at least potentially) similar power and status.
Does this help?

Looking for an actual play example of how a duel plays out. An instructional video, an online game where a duel took place, a blog post that gets a lil more descriptive than the rulebook, etc.
Is there anything like that floating around?