Just used the Thunderbolts with the Serpent Society-Can Use Phasing (Even If It's Not on the Dial)- and kicked butt. Bullseye with Phasing and running shot was what won me the game. What other team abilities should i use (other than mystics)?

I am pretty sure phasing is just on a move action. I also believe that Serpent society TA is only available via ATA now.

yes but running shot says "gives his character a move action up to half their move value, this character may then make a ranged attack as a free action"

Actually it says give the character a power action and move up to half it's speed value. So unfortunately it doesn't work together. Also, serpent society is an ata now and can't be thunderbolted to.