Disclaimer, I've only run one game of FAE, but I spent a lot of time beforehand reading the rules and listening to people play.

For your first bullet point about checks, I'd say this is really all about GM and group style. This is really something that happens in any game. I like to run by the guideline of having players make checks for things that I have interesting fail states for. If I can't think of something fun or story provoking that might happen if they fail, I just let them succeed. But I've also played with groups that wanted to run checks for every little thing.

As an aside I will say that I found asking players for checks specifically to be important for helping players pick up on when to use what approaches. Rather than only making skill checks something like "here is a door, how do you get through or around it" I would also do things like "give me a Careful check as you cross this rope bridge." Even if they back up want to use a different approach to cross the chasm a different way, you've given them a solid example of what kind of checks Careful is good for. Alternately if they don't mind making the Careful check they can just roll quickly and you can move on without a long discussion about which approach to use and arguments about how a player could use their best approach.

For your second bullet point about compels, I found this one of the harder things to pin down. I ultimately didn't make my adventure hard enough so my players weren't really ever lacking in Fate points. I did use compels to try to maneuver players around into places that would get them in trouble. My group is usually pretty good about picking up on a plot and following it, but I think compels are the most useful when your players are arguing about how to get through a situation and you can find a way to compel the player into action.

Lastly, I wouldn't say that the rules are ambiguous as much as I would say they are 'light.' It isn't a system like D&D where if a player wants to use their tied up hands to throw soup in the eyes of their jail keeper and you can flip to the page in the DMG for improvised weapons and throw on an impediment negative to their roll. The rules in Fate are more like a light touch set of tools that can apply to any situation. This means that for any situation you could apply the rules in different ways to get the same result. If your player is having an argument with a professor and they want to jump down their throat and really start yelling, you might want to set that up as a contest to see who wins the fight while I might have the player roll one opposed Forceful roll with the understanding that its socially forceful not physically, so any fallout from failing the roll would be mental.

Does that help?