Darkest Dungeon

What path can I choose when resolving this ability?

See Moving to a Specific Location Via a Special Ability or Magic in the Codex.

What happens if there's no path that can get both the Avatar and the striking unit to the Darkest Dungeon (because of Perilous Bridge, for example)?

If there is no legal path that will get both of the involved units to the Dungeon, there will be no movement put on the storyline, and both units will remain where they are.

What happens if the ally strikes the avatar during a fight, and is killed by the avatar's strike?

If the ally dies as a result of the fight, then you are unable to drag both the ally and the avatar to the Dungeon. The avatar will not .