I have 5 Pokemon on my Bench, and 2 of them have the "Omega Barrier" Ancient Trait. If my opponent plays Avery and I choose the 2 Pokemon that have "Omega Barrier", do they get discarded or is the effect of Avery blocked?
Yes, they get discarded. The discard effect of Avery is on you, the player choosing the Pokemon, not on the Pokemon with Omega Barrier.
Source: TPCi Rules Team (2022-01-20)
If I have Milotic with the "Dew Guard" Ability in play and my opponent plays Avery, do I have to discard Pokemon from my bench until I have 3 or not?
Milotic's Dew Guard prevents all effects of your opponent's Supporters that are done to you, so you would not discard any Pokemon from your bench for the second part of Avery.
Source: TPCi Rules Team (2021-09-09)