They will yes, eventually. I don’t think it will be for a little while yet tho. Round 2 of costumes only started in Beta after the first round of costumes were all featured.
So it could be 3-4 more months before we see new costumes in Beta & hence into live game.
I personally expect sometime in August or September we’ll start seeing them. Make sure you subscribe to beta-beat to get a notification of new Beta Content 
Not going to happen I don’t think… SGG have designed it this way & haven’t made any indication that it isn’t working successfully. Feel free to vote for it as a Ideas & Feature Requests tho here: [Master] Costumed heroes should be able to use emblems from their new class