Can I use cards that search for a Pokemon, like Ultra Ball or Poke Ball, to get a Pokemon V-UNION card from my deck or discard pile?
Any effect that lets you search your deck or discard pile for "a Pokemon" will let you use that effect on a Pokemon V-UNION card; but remember, you cannot put a single Pokemon V-UNION card into play.
If I use Medicham V's "Yoga Loop" attack to KO one of my opponent’s Pokemon, per the effect I get to take another new turn. If I then play Pokemon Ranger, can I use "Yoga Loop" again?
No, Pokemon Ranger won't bypass the repeat limitation of Yoga Loop because that is not an Attack Effect, it is a Game State Check. Yoga Loop is checking what the game state was in the past (did you use it last turn?), and Pokemon Ranger has no effect on that.
Boost Shake says it can be used during my first turn or on a Pokemon that was put into play this turn. What if my Basic Pokemon was in play last turn but I evolve it into a Stage 1 this turn; can I use Boost Shake on that Pokemon to search for a Stage 2 and put it onto that Pokemon?
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.