Is the new Masters of Evil team ability legal in tournaments now, or when Sinister is released? Also, does it allow for repeated pushing, or just one 'extra' push?

It will be part of the new rules set released with the Danger Room starter (Sinister really doesn't have anything to do with it, DR just happens to be coming out the same time as Sinister ). The rules changes won't go into effect until the release; sometime around July 12th, I believe.

It appears as if this ability works just like the Sentinel/Sinestro figures do. You can continually take actions while continuing to take pushing damage.

Actually, it isn't pushing damage, it's unavoidable damage. This is an important distinction, as WP wouldn't reduce unavoidable damage.

If it was pushing damage Baron Zemo would just rule the roost