In the article "Prepare for Ground Assault", Elaine's X-Wing chooses to hit the TIE Fighter because "it would only scratch the Star Destroyer." This caused me to wonder why that wouldn't be worth it, as 1 extra point of damage later on might be very useful.

So, does the damage heal at the end of the battle, or the end of the phase in which it takes damage, or do ships not heal at all?

Damage does heal at the end of the battle.

In that example, even tho the xwing could have damaged the SD, it went for the kill on the Tie Fighter since it was dieing anyway.