I'd say work on E1 everyone you use, then go from there. E2 is very expensive and only a few characters really benefit from it as of right now (Unless the new chapter has changed things). The general rule of thumb is double up on each class (two tanks, vanguard, etc.) FEater, Mayer, and Cliffheart are situational and are not necessary to level up unless that's what you want to work with. Cliffheart's and Shaw's ability is very strong at lvl 4 even when they themselves are lvl 1. I've cleared the 2nd annihilation using a lvl1 shaw and cliffheart and SK-5 using lvl 1 shaw.
Pretty much this. Just focus on getting your "usual" team to at least E1 and then go from there.
The only thing I would add to this is rather than doubling up on each "class", double up on each "role". There are characters who are technically of the same "class", but fill a very different role.
For instance, I use Melantha on a lot of maps as a high-hp/"defensive" guard. That same role can't be filled by Midnight or Beehunter, even though they are both guards.