Do I have to have the actual character card with me to use a special power?

Yes. Although some judges have said that they have made a printout of ALL SPs to use for reference, so their players wouldn't necessarily need the card to use the SP.

Would you trust someone who said, "Well, Iron Man gets to shoot you, I think. I can't remember if it is a power action or a ranged combat action, though... Hmm, let's just call it a power action. Now I get to roll one die and, umm, do I add that to his damage? And I take a click... No, I take 2 clicks. I think."

Just like feats, you need to have the cards in front of you. Also for keywords, too.

As a judge, if a character had no special powers and the player wasn't trying to run a theme, I probably wouldn't care too much if he didn't have the card. But as a general rule, I'll try and make everyone get in the habit of bringing them anyway.

Ditto that. I won't bring my cards if I'm not running a theme team and I won't use the SPs (or if the fig doesn't have any). Why bother if you won't use them?

Because if an opponent MIND CONTROLS a figure with a special power, the opponent should have the option to use that special power (and know what it does).

It would be very fun to Mind Control the new Ultimate Iron Man and do a Full Power Energy Explosion - 2 feedback damage to IM and no extra damage...

Or have him Full Power one of his Mystic Allies!

Ooouuuch! (had to be 10 charachters)

Well, in that case, it's best just to cancel your Mystic's TA.... and hope it wasn't a ruse to get your Mystics to drop his TA so that the rest of your opponent's team can smack him without retribution.... that always sucks.

Wouldn't you have to turn it off on your own turn? Or can you just turn it off like that?

From the LoSH Rulebook (p. 15): "All team abilities are optional and may be canceled. If a team ability is not canceled at the beginning of an action, it is in effect during the action. When a team ability is canceled, it remains canceled until the end of the turn and resumes effect at the beginning of the next player’s turn."

You can cancel your TA at the begining of an action, even if it's during your opponent's turn.

IIRC Hair10 had ruled that the "mind control" action constituted a new opportunity for characters to switch up/off on Team Abilities. I don't like it (only because I think the rules are already too liberal on TA changes) but it would be consistent with other rulings.