Is attacking population and influencing hexes done at the end of battle, or at the end of the combat phase?
I would say option 3. That seems to follow the Course of Battle as detailed on page 20-21:


Missiles
Engagement Round(s)
Retreat
Stalemate
Attacking Population
Reputation Tiles
Influence Hexes
Repair Damage
Player Elimination


Where 1 through 4 occur for every battle. Step 5 through 9 occur after all battles across the board are resolved with the exception being step 6, since drawing reputation tiles occurs after every battle.

Attacking population and influence hexes doesn't necessarily have to be done in descending hex order.