What would happen if I touch two different pieces with each hand at same time? For example, I touch my king with the left hand and my queen with the right hand at the same time. Regarding the rules, I must move a touched piece, so what would happen in this case? Could I choose which piece to move? Would I lose the game automatically when doing such an action?

Is using both hands allowed at all? (e.g.: moving one piece with both hands).

As far as I know, at least in blitz, you must use the clock and move the pieces with the same hand, so what would happen there if you use both hands when moving a piece or touching two different pieces with each hand?
From the FIDE Laws of Chess, article §4.1:


  Each move must be made with one hand only.


So you are not allowed to make a move with both hands (this includes castling) and the arbiter can give you a penalty (§12.9) for this behaviour. IMHO, this should be a warning the first time you do it. In any case, because you touched the king and queen, you must make a move with either piece (§4.3):


  If the player having the move touches (...) one or more of his own pieces, he must move the first piece touched that can be moved.