Do 2nd stage evolutions still give you the CD move during a CD when you evolve it?

Evolving from a Stage 1 (like Croconaw) to a Stage 2 (like Feralgatr) will give you the CD move (like Hydro Cannon) as long as it's done during the CD move time (typically 3 hours of CD + 1 hour of evolve time).

You can only get the special move for the pokemon family that is the focus of the event. For example, the upcoming CD is Totodile and you can get Hydro Cannon on Feraligatr that you catch or evolve during the 3 hour event and up to 1 hour after.

December's CD was an exception. It was a repeat of all the previous CD pokemon from 2018. Any of those pokemon could get their special moves during the December CD, but will not be able to get them during this month's CD.

-------

I think you can only create one new trainer name. But I think you can also choose to switch back to your first name. I haven't done either, so can't say for certain.