Questions about the French Defence

I've played the french fairly often and the f-pawn break doesn't usually come up too often for me, unlike the c-pawn break. Either capture has its advantages (capturing with the g-pawn allows additional breaks in the center), so you have to judge based on the position. Take a look through some games by master to get a good sense for it.
As far as castling, I tend not to castle often. Castling queenside is almost always a bad idea since the c-pawn break leaves the queenside too exposed for the king. The center is usually closed for a long time with the french, so castling is not always necessary, but watch out for the center to suddenly open up on you. In my games, by the time the board naturally opens up, there has been enough trades that the king isn't in much danger anymore and is well placed in the center.
A queenside side attack with a6 followed by b5 is a possible way of attacking in the french, especially in the advance variation. I've had fairly good success with it, but it can be a little slow and its possible to become overwhelmed by white on the kingside; but that's the french in general. The most important theme for black in the french is the undermining of white's d4 pawn, especially in the classical and advance french.
That's my two cents anyway; hope that's helpful.

The idea is that after you recapture with the knight the backward pawn is offset by the open f-file and the open c-file. Its up to you whether this is enough compensation for the positional concession or not

In the winawer, the recapture ...gxf6 is common, if you're going to castle queenside. It opens the g file and allows ...e5 to be played, and black usually wants to close queenside files with ...c4. But most of the time a piece recapture is more normal and solid; it all depends on the specific position though. I would disagree though that castling queenside is always bad in the french, because many times even in the advance variation black goes queenside, with the idea of recapturing on f6 with the g pawn or to play ...g5 and ...g4 to attack a knight and indirectly pressure the center quite a bit. Black just has to keep his c pawn on c5 or push it to c4 and he's often fine. If he has played ...cxd4 totally opening the c file, then it's usually bad. You know, it's a complex opening, and for the french general guidelines probably won't give you an understanding of the position. You should get a french defense book to know the ideas better so that you can find out what's correct yourself with the new knowledge.
For example, black's goal is often to remove the center pawns, but even if he does, it will only be good if his center pawns are mobile. If not, then they give white central squares on d4 and e5 (basically replacing the pawns). So it's double edged and all depends.

It depends on what line of the french you are playing. I can't think of any where the g pawn is better its most often the knight captures pawn then the bishop goes out then castle 0-0
I always see where you want to castle but the kingside is too dangerous to castle. From what I've read sometimes its better to wait. Others even recommend castling 0-0-0.
Recommended Reading: New in chess Moskalenko The Flexible French or Everymans french classical.