Can you add to or remove from fleets literally whenever?
Pretty much, as long as it's during your command phase and not in mid-move.
Can you reorganize like this before the start of a space battle?
If so, who has to finalize their fleet organization first?
Has to be during your turn; only the attacker may do it.
How about between combat rounds, perhaps to reinforce one fleet with the remnants of a second fleet that was disbanded due to damage?
No, can't be done mid-battle.