Guzzlord

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 opponent uses Guzzlord's "Red Banquet" attack and my Active Pokemon with Big Parasol attached gets KO'd, does my opponent get to take an extra prize for "Red Banquet" or not?

Big Parasol prevents all effects of attacks from your opponent's Pokemon done to your Pokemon; but taking an extra prize is a game effect, not "an effect done to your Pokemon", so your opponent gets to take the extra prize.