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Barring special situations (Second Seal/Siege, The Darkness), you can't add a character to an opponent's side of a battle.

If you could do it, whoever controlled the side of battle it was added to would control it.

I haven't done a of research into why but I've always heard that you can't add character to an opponents side of battle. The whole at any time clause isn't relevant if by ruling you cant add a character to an opponents side of battle barring a special ability that specifically can. Card's in general aren't usually aligned or worded to allow that in the first place but I'm sure there are some corner cases and exceptions so I can't speak too much as for why. If specific cards and abilities were mentioned it might be easier to talk about but again I'm not the most knowledgeable person for that.

However most of your questions can be answered by reading the REG entry for Add to Battle. This section covers most of it.

Default Conditions●Targets must be in a territory or in holder’s hand.●Add to battle abilities target cards of the same alignment.●Add to battle abilities can only add cards to a battle in which you are participating.●Adding a card to battle causes the player on that side of the battle to control the card. If a different player normally has Permanent Control of the card, the player on that side of the battle gains Temporary Control until the card leaves battle

Hopefully that answers some of your questions at least.

I believe this old thread is the most relevant that I could find

http://www.cactusforums.com/ruling-questions/add-to-battle-clarification/msg573463/#msg573463