Hearts of Iron III: Or how I managed to start WWII

[Grif]

So... I just gotten this game as part of the Steam's Paradox weekend. (Up to For The Motherland) I'm somewhat sad/happy to say that I'm now addicted to this game.

That being said, even after playing for hours on end (as Poland, because trying to hold out against the German tide is just fun), there're still some features which I cannot wrap my head around. Perhaps those of you who are more experienced than me could shed some light.

How is fuel actually converted and what determines the conversion?
Is building militia or garrison brigades useful compared to regular infantry?
How the heck does AA work in this game?
Any solution to the disappearing menu background glitch? It seems to occur when a popup occurs while I'm browsing another screen that is not the map.

EDIT: How could I forget the most basic of all questions: What mods should I get to make it even better?

[Gwyn chan 'r Gwyll]

I don't have For the Motherland, but I'm still basically familiar with how things work.

Garrisons are a cheap way to, well, garrison key points that you don't want to stick a whole proper unit on. Ports and airfields, usually.
Militia are basically a stop-gap, if you can't afford to field enough infantry.
Fuel is converted automatically. There's a tech line that increases the rate of conversion. Other than that, it's handsoff.
AA fires whenever enemy aircraft is detected in the province it's in. Same with mobile AA.
I dunno about the UI glitch, and I don't know which mods are worth it...