Just checking:

If Barrier power is canceled, countered or punched away, does a prior materialized barrier also disappear?

If so, does this also apply to outwit. Other powers?

Once a barrier is up, it stays up until the player's next turn, regardless of what happens to the character that made the barrier.

Actually, if the power is countered or lost, the barrier does disappear.

From the updated Indy PAC:

BARRIER (optional): Give this character a power action. This character may create up to 4 squares of blocking terrain anywhere within its range. Place up to four barrier terrain markers on the battlefield. Each barrier terrain marker must be adjacent to another barrier terrain marker created by this character. These terrain markers remain in place until the beginning of your next turn or until the power is countered or lost.

So what if someone punches/shoots a hole in barrier. Is hindering terrian left behind for that turn or do you just take that part away?

I would assume it just dissappers since it is just a Force Field/wall of fire/ice.

and can you fly/soar over a wall or shot over a wall if you are on elevated terrain?

A wall? Yes. A Barrier? No. They are considered "infinately high".

SO you can't shot over a Barrier or fly over one. Also a destoryed barrier becomes hindering terrian until the barrier is gone.

I was wondering if this info is on any of the wizkids faqs.

Like Psylockeslover just said, on an outdoor map you may fly over a barrier.