Team Rocket's Koffing

If I use Team Rocket's Larvitar's "Mountain Munch" attack against my opponent's Team Rocket's Koffing with "Smog Signals" Ability, do they discard the top card of their deck for "Mountain Munch" first or do they get to search using "Smog Signals" first?

The step to check for effects that activate when a Pokemon is damaged by an attack occurs after the step to check for effects other than damage. So the attack effect of discarding their top card is done first, then the triggered effect of searching is done afterwards.

If my Team Rocket's Weezing is damaged by an attack, do I have to use "Smog Signals" to search my deck even if I don't want to or if my Bench is full?

The use of the Ability is not optional, so if you have bench space available you must search your deck and shuffle it afterwards. Of course you can voluntarily fail the deck search, but you must search and shuffle your deck. If your bench is full, then you cannot search as there is no place to put a Weezing if you did get one.

If my opponent's Active Pokemon is Cetitan ex with the "Snow Camouflage" Ability, can I use Team Rocket's Giovanni to switch one of their benched Pokemon with it?

If your opponent's chosen Benched Pokemon is Cetitan ex, then Team Rocket's Giovanni's effect will be blocked. But choosing a Benched Pokemon without "Snow Camouflage" will cause it to switch with the Active Cetitan ex because the Benched Pokemon is the target of Team Rocket's Giovanni. You will still switch your Team Rocket Pokemon either way.

If my only Pokemon is Team Rocket's Koffing with "Smog Signals" Ability and it is KO'd by damage, do I get to search for 2 Koffing and put them on my Bench or do I lose the game?

Yes, you get to use the Ability before you lose. Team Rocket's Koffing takes the damage and you search your deck for 2 Pokemon with Koffing in their names and put them onto your Bench, then you check for KO's afterward.