castling or not?

The King is normally safer behind the fianchetto than behind 3 pawns. But if the fianchetto Bishop is traded, then the King position is weakened.

I'm assuming you mean a Kingside Fianchetto. Yes you should castle, but if the opponent can get that Fianchetto off the board, your position will be weakened. That's why in some cases, the Fianchetto would be worth more than a rook. Even though technically the bishop is worth less than a rook, the Fianchetto is extremely important to the defense of the king. 

Important to consider if your opponent still has the Bishop that counters your fianchettoed one...if that Bishop is gone, the weakness of the g2 (or g7 if Black) square is much more difficult for your opponent to exploit.

Never trade off a fianchettoed bishop that protects your king unless the opponent's same colored bishop and queen are also off the board. For example, if you're white, castled kingside, and fianchettoed your bishop on g2 (a light square), don't trade off that bishop unless the opponent's light squared bishop and queen are also off the board-- you don't want to give them access to the fianchettoed pawn structure.

"If you have your bishop in fianchetto do you castling or not?" ++ Yes
"Is castling behind a fianchetto as safe as castling behind 3 pawns?" ++ No, the fianchetto is not as safe, as it provides attack marks for h2-h4-h5 or f2-f4-f5 ot Be3 Qd2 Bh6. The power of the fianchetto bishop which runs through 2 central squares makes up for the weakening of the squares around the king of the color of the fianchetto bishop. The fianchetto bishop defends the suares weakened by the fianchetto.

Some people seems to overestimate the Kings safety behind 3 pawns. There are chances to open the position with sacrifices. The Fianchetto make this more difficult, but the pawn on g3/g6/b3/b6 surely make an a- or h-pawn rush (like AlphaZero or Larsen) more dangerous. The point could be to take and open line, but it could also be to advance to h3/h6/a3/a6 and restrict the King + have an strong pawn in an endgame.

I think #4 and #6 are good remarks. Castling behind your fianchetto is safe. That being said, you do weaken your king's position as the pawn you push forward can be used as a hook to attack. It'll depend on the position whether this is a problem or not. If there is opposite castling, pawn storms against the enemy king become very normal. Having a hook to target means that lines will open up more quickly than if you didn't have one. That also means that the response needs to be just as quick, so you'll get a game where no one can afford to make an inefficient move because the pace of the game is so quick. In case of same side castling, I do think the fianchetto will be hard to break down, but absolutely not impossible. Look at the Grand Prix attack against the Sicilian for example. 

Even though you usually want to keep the fianchettoed bishop, trading the fianchettoed bishop for it's opposite part can be OK, especially if there already have been some trades. Usually you'll use your king to defend the weaknesses that remain. For example: Kg7 defends h6 after the fianchettoed bishop gets traded (on g7: we never initiate ourselves). 

Please note that castling queenside after fianchettoing there is comparatively rare and I would feel a little bit less secure. To feel completely secure you'd need to move the king over towards to b1/b8, but that might lose a tempo that could be costly if there were to be a quick paced attack that uses that fiancetto-pawn as a hook.