A character has both feats assigned to it.

Would LMD still be in effect, even though the character is KO'ed and off the board (though still technically in the game)?

Player's Guide:
"Characters that are removed from the battle map (but not
eliminated) are not affected by game effects that characters
on the battle map possess (unless the effect removing them
from the battlefield specifically says that they are).
Characters that are removed from the battle map (but not
eliminated) may not use their own powers, team abilities,
or feats except as stipulated by the effect removing them
from the battlefield."

What constitutes "using" LMD? I would personally rule that any effect the card has positive or negative is considered to be using the feat. (I guess you could call it actively or passively using the thing.)

I would say that you could be healed.

Ah, thanks! I couldn't remember where to find that text regarding feats when off the map and wondered if I actually had read it.