Well... this is not how the game was designed to be played. I see no problem in use a house rule, but I think that if you remove the non destroyed UFOs the game will become much easier.
 
The game "lends" a lot from the videogame but it is not the videogame. For example, I'd like that soldiers took advantage of killing aliens to gain ranks or become elite, maybe even a diferente power or 1 extra "hit point". It does not exist in the board game but may exist in a house rule. The main question here is how all these house rules will take effect at the game balance.

This has already been "solved"; all invasion plans are 7 rounds long, except for Onslaught which is 6.  Each normal mission you complete reduces the number of rounds until the Final Mission appears by 1.

I have lost the tutorial twice, and won on both easy and normal.  I believe the tutorial is weighted to knock you out once it's taught you how to play so you can start your own game.  Both the games I won ended in the sixth round.

The tutorials seems to be very difficult. We tried the game on easy and got managed to win though. 

Kinda thematic - they obviously died preventing the other crisis

@OP: Well, let's be honest, if one were to hit the menu button to extend the timer, one might just as well turn the dice to whatever result you want while you're at it.
The best way to move on to normal without needing extended pause time is to figure a way to streamline your turns. Some tips if you're interested:
1. Separate your starting assets and tech cards into Phase groups. At the end of the resolution phase, I put all my resolution tech and asset cards in a pile off to the side, because I know I won't need them until the next resolution phase.
2. Get to know the invasion plans, and what tech is most helpful for those plans. For example, Onslaught presents tougher enemies and an almost always fully loaded base, but very few UFOs until the Final Mission is revealed, when suddenly they seem to flood the board. So you can make use of early research that uses up satellites to do cool stuff, since you won't have too many UFOs that need to be moved around with your Satellite Nexus until the end. You can probably do without cards like Firestorm (although it's never bad.). Instead, research the weapon tech and anything that helps keep your soldiers alive.
3. Layout your timed cards in front of you, in order of general importance. Satellite Nexus, Skyranger and Officer Training are usually the most important.
4. Keeping the budget in mind is certainly one of the toughest aspects for me in single player. I solved this by keeping the number "12" in my head every turn. That's 3 scientists, 3 satellites, 3 interceptors, and 3 soldiers to defend the base. If you want to do anything beyond that, it's gonna cost more than 12.
5. Vacuum up UFOs into orbit whenever you can with Satellite Nexus and fight them there. Always roll scientists and satellites to exhaustion. For Interceptors, roll twice and stop unless things are dire and the situation is grim, or you've got Interceptor Repair. If you don't, the buy-back on planes can bankrupt you, especially if you're losing soldiers.
6. Once the enemy deck is put together, look through it before you shuffle. Figure out how many of each icon have yellow borders, as this will tell you which soldiers are going to be doing the heavy lifting. This helps you to know which units need to be made into Elites. This also helps with knowing what techs to research. For example, if you know your Assaults aren't as important in this game, you can research Run and Gun, and keep your Assaults back so that you can use it.
I could go on and on and on, but this wall of text is too big already.

First things first.  I love this game. Tense, fast moving, unexpectedly deep. My issue is this.
 
I have no other people around me that are into games - not even the wife.  So my option is solo play. So far I'm finding the easy level 'pretty challenging' for being 'easy.'  No complaints there, I'm apparently a slow learner. 
 
I'm getting better and at some point I'm going to want/need to play at a higher level. That will be impossible for a solo player. Difficulty rising, time still running, it just isn't going to happen. 
 
I think that it would be nice if the development folks would allow solo players to continue to pause the game to take a few more seconds to figure the next move. So if anyone is listening; some of us, I can't be the only one, would like to eventually be able to play the upper levels solo. 
 
That said; am I right in understanding that hitting the menu button is like hitting the pause button? If so, then I guess that makes the whole point I'm  trying to make a little 'pointless'.  
 
That assumption about the menu  button is correct; isn't it?
 
Thanx, 
 
dizpatcher

Like Bleached said, the UFOs don't go anywhere just because you haven't killed them.  The "destroy 2 UFOs" just makes it easier to handle global defense next round if you've left some UFOs still on the board.  I have found this is typically the least useful mission reward, but it can be great in the right circumstances.
 
I actually really like the order of the resolution phase.  Every portion of it, excluding science research, can raise a country's panic levels or do damage to the base, all of which push your "loss" condition of the game.  You have to survive all of these steps before reaching the missions - especially the final mission, which is the only way to win the game.  I like how you can't just abandon something completely and just push for the final mission.

UFOs stay on the board.  The point of the mission rewards is that those UFOs won't be there to raise panic again *next* turn.

Answer to question 1 made by MyNameIsDonovan is wrong for many reasons. If that is an official answer from a designer, well I'm sorry but shame on him...
 
First of all, game has official terms so let's use them: tech cards. Research card is something people don't know. They might try to guess, but we are here to clarify not to mess everything up more than it already is...
 
Now second point is that TECH CARDS MUST ABSOLUTELY NOT be placed on the bottom of the deck! Just play the game once and you'll understand why...
 
The Chief Scientist may play Alien Alloys and "Place 1 discarded tech card on top of the tech deck". Now if we've placed the discarded tech cards on the bottom of the deck the scientist is not able able to select the card among the discarded ones. He/She can only select the card among the entire deck instead. Even worse, in this way the scientist has seen the sequence of the entire deck and thus the deck needs a reshuffle (we are fair players, right? don't cheat please!). So keep a side deck of discarded tech cards, reshuffle the side deck only when the main deck is out of cards. This works fine and allows you to do exactly what Alien Alloys tells you to do.
 
Otherwise if we want to say Lukas was right, Alien Alloys card must be wrong and needs at least to be rephrased.

Unless they die, I'm pretty sure no one should be leaving the board during a task.  They are removed during the refresh phase.   If you have 3 enemies to fight and 4 guys assigned to base defense, you can assign some combination of troops to the first enemy, then, you could reassign a different combination of troops to that enemy if you hadn't killed it yet.   The threat tracker still goes up for each role, but you can change who you want to risk loosing.  

I'm afraid yes you do lose all interceptors in a continent when you lose a die roll (same with scientists and satellites).  It's an encouragement not to put all your eggs in one basket. 
 
On the first turn of a game I lost 3 soldiers on base defence then lost another 3 on a mission by rolling 1 twice in a row which sucked balls.  I did still win the game (solo).