If a figure has say....Mystics, and has Lazarus Pit, and is KOed, is their TA still available for wildcarding?

Not while they are off the board. This is similar to Oracle on her card; if the character isn't actually on the map, you can't copy its TA. Once the LP character is back, though, copy away.

And if you copied it before they got swacked into the Pit, your figures retain that TA now, as I am sure you are aware...

I can't look at the card right now. Until the 5 or 6 is rolled for the returned character, is its TA ignored? cancelled?

It can't use its TA... but that is not to say that the TA isn't there for others to copy.

Kind of like when only one X-man is left on the board; it still has its TA, but there is no way it can use it.

Okay, just to make sure I'm on the right page:

Mystics character has Lazarus Pit, gets KOed, ends up on the card.

Wildcards can't copy it.

Mystics character comes out of Lazarus Pit, doesn't roll well, and retains it's lazarus pit powers. It can't use Mystics, but everyone else can.

Yes. Since you can never counter a TA, the LP character still has a TA on its dial; it just can use it while under the influence of LP. Meanwhile, as far as your Wildcards are concerned, there is a character on the team with a copiable TA and they can copy it as they see fit.

Look at it like this: RCE is a prerequisite for Trick Shot, but you don't need to actually use RCE to use Trick Shot (like during a Running Shot action). However, if your opponent Outwits (counters) RCE, then Trick Shot can't be used. So, think of the Lazarus induced 'rage' as a Running Shot and the character's TA as RCE; it can't use its TA, but other Wildcards (think of them as Trick Shot) can use it since it is still there on the combat dial.