I believe the wording in the new rules for the Skrull team ability no longer includes "Using shape change in this way does not prevent the character from using shape change through another game effect"; so my question is... Can you still use both the Skrull TA and the power of shape change on the same character when he is attacked?

Also, if my character A has the Fantastic Four printed TA, and I also have character B with a wildcard TA copying the FF TA, when character A is KO'd, does character B still get a click of healing?

You are correct, it no longer appears in the TA so a figure with Skrulls and SC is not so great anymore.

Yes. "FANTASTIC FOUR-When a character using the Fantastic Four team ability is KO’d, each other friendly character using the Fantastic Four team ability may be healed of 1 damage. If all remaining characters using the team ability are KO’d as par t of the same action, this team ability can’t be used."
(Emphasis is mine)

Thank you.