Now that we have a phasing TA, if I have my (V) Dr. Doom copying the SS TA, am I able to use RS with it?

My gut instinct says no because Phasing and RS are both Power Actions. And we all know that only one Power Action can be assigned to one character at a time.

Yep, you're gut instinct is right on the mark.

The only thing that is still possible at the moment, because the wording hasn't been fixed for the TA is that you get an automatic break away with the SerSoc TA, even if you're not using it. This is probably a copy+paste mistake, which has been carried over from the wording of Phasing. So in principle you can do an auto-break-away RS with Doom at the moment, but you can't phase through a wall while performing a RS. But hopefully this loophole will be closed with the next FAQ!

Well, since when do "continuous" team abilities require activation? Catgirl would still be stealthed while she did a running shot, assuming she was behind terrain, Hawkeye still gets to be Ultimate when he does his running shot.

I think they would work together, and it would add a new, messed up dimension to game play, like the new Thor.

Serpent Society is not an continuous TA, which is always on. The wording specifically requires the expenditure of a Power Action to use the TA. Exactly like the use of Phasing, which is the reason why Phasing can't be used in combination with the Avengers/Brotherhood/JLA/Top Cow TAs, which only make Move Actions not count towards your total action allotment for the turn. I.E. Vision can't phase "for free".