Wasp

If you push a unit into terrain or another unit, does it take damage? Or is there a specific hero edge case where this happens?

In general- no. Wasp can cause discard that way though.

Can a hero inside Wasp's barrier be attacked in melee by a hero from outside the barrier? Trinkets' Turret is an obstacle and cannot be attacked, and Wasp's barrier makes a hero inside it an obstacle for heroes outside — does that mean the hero inside is not a unit?

1. Yes, why not. 2. It's not important if it's an obstacle or not. What is important is that nothing says that it's a unit. 3. Wasp's barrier does nothing to heroes, only to spaces.

Wasp's silver says spaces count as obstacles, and Bear cannot swap with an obstacle. Does Wasp's silver prevent swapping?

Wasp’s silver doesn’t affect swapping.

Can Wasp and Tiger Claw block a ranged attack if it is used to target only adjacent targets — e.g. Dodger's Finger of Death?

“Ranged subtype“ paragraph has the direct answer to your question.