Can my opponent choose a different one of their characters for each 1 damage? Can I choose a different creature for each 1 repair?
[Source: WotC]
Yes to both.
Not right away. Set them aside for now, in the same place you put played Actions cards and battle cards flipped during the attack. Those cards will all be scrapped at the end of the turn.
Are cards flipped during battle the only ones that don't hit the Scrap Pile until the end of a turn?
[Source: WotC]
Actions that are played (including Secret Actions), cards you flipped during battle, and other cards you flip due to cards like Swarm!, for example, are all scrapped at the end of turn.
Yes.
Junkheap's bot mode says "When this defends and your opponent flips battle cards", allowing its owner to scrap one of the battle cards flipped. What if the attacker attacks with
Anticipation Engine and plays Swarm! from the top of their deck; can the defending Junkheap player scrap one of those cards?
[Source: Aequitas]
Yes, they can choose to scrap one of the flipped Swarm! cards because it is happening within the battle while Junkheap is defending.