1. Pre-Exodus, yes, I'd agree nukes don't seem too valuable if a jump is feasible. They are probably more valuable in Exodus though as ships not destroyed will come back to haunt you. In Exodus, you can build more nukes though, so I suspect we will see more nuke action in the future.
2. It is 1, 2, 3 in sequence and yes it is pretty common that there are no ships to activate when it card is revealed.
3. I don't think the designers intended all human loss conditions to be equal. It just means you can't neglect any of them. Fixing Galactica or killing centurions isn't too hard to prevent the loss, but those are actions and resources spent that could be helping other aspects for the humans. Most cylon victories are on the resource dials for us.
4. The resource is lost at the end of the jump, but if a dial ticks to zero at the moment of victory it is still a loss for the humans. Might not be how you thematically envision it, but I'm pretty certain that is the rule. I guess, picture it as a tragic finish for the humans as they see their final escape, but the fuel depletes and they are unable to complete their jump.
5. Most (if not all) of the if the "you are a cylon" cards lose their special bonus while brigged, so if you have the resources to spare (action/skill cards), a good tactic is to contain a cylon before they reveal. Unfortunately in BSG it is often hard to be sure of who's on your team or not unless you've seen their loyalty cards and then you still have to convince the other humans while likely facing accusations yourself. Good times.
A revealed cylon cannot be brigged.
If you get the Pegasus expansion, it gets a bit confusing, as a cylon leader can be brigged while infiltrating and they have conflicting victory conditions (e.g. humans win but with 2 or less of a particular resource remaining). As a result, they often both hurt and help the humans while trying to meet their agenda (this is also true for sympathizers in Pegasus). So if you fear them being too damaging brigging them might be a good move, but we've had it hurt the humans as well.
6. In the base game, the sympathizer works as a balancer based on the humans current resource status when the sleeper phase is triggered. If they are brigged and helping the humans, as a human you would prefer them out of the brig to help you, but only if you can spare the cards. The consensus seemed that being the sympathizer was a damper on the fun, so in Pegasus they have them receive a cylon leader agenda card, so they at least get an interesting game goal to work for and it makes their intentions more mysterious and interesting.
Glad you've enjoyed the game so far, it is an amazing social board game that we love. Like the show, the game designers intended for the human struggle to be very challenging and when they win to feel pretty epic. The fact you have a 50/50 ratio says to me that your cylons need to start bringing their A game. Easily in our games, most of our victories go to the cylons, but we often have some very crafty players who make being a human extremely challenging and thus we often get distracted from immediate dangers while accusing and being accused.