One useful technique for avoiding assassinations is to take the snake as your pet so that you can set your mood to indulgent and then move it to social. I've never seen assassins on the "social" mood, and I did a run where I was specifically trying to bait assassination attempts (that Zorana survived 10+ assassinations and got 8 electors disqualified from voting lol). That said, it's possible to go through the whole game without facing a single assassination attempt if you do a great job at keeping everyone happy. Saving at least once a turn is definitely very helpful, though. When you fail a check the game tells you what you needed to pass it, and often if you go back a turn or 2 you can train those skills.
Unless you're going for specific scenes that require a very high value in a certain stat, I've found that the knowledge skills aren't very helpful to train, it's easier to just get those stats from your cabinet. There's not really any single stat that's a "must-have", though, most checks can be passed in several different ways. There are a lot of checks for empathy, but most of them are just for telling whether or not someone likes you, not actually influencing people's opinions. Rhetoric and especially etiquette also have a lot of checks, but many of them have pretty low requirements. I've found that just grabbing student level (30) in both is generally enough to keep from offending everybody.
Personally, I've never recruited the admiral via spaceships (yet). Having "survived an assassination attempt" (which you can get for free by having the Thresher sleep in your room), "played ritual Starfall" (which I already like to do to recruit Ethexae) and the wrestling accomplishment from either Lorkas' or Radomir's quest is enough ti impress him.
I don't know the exact mechanics behind assassins, but the thing I've found helpful in picking a cabinet is making sure that none of the members are enemies with each other, because that was explicitly said an update to be something that tanks their approval of you. Having a high skilled cabinet doesn't inherently increase assassination attempts, it's just that some of the electors with the highest stats also have a bunch of enemies. Also, if you haven't found out, getting intrigue to 100 once is pretty useful for figuring out exactly what each elector has to offer you.