If it were me I would not bother insulating in NC unless you are on the top of a mountaintop (and you wouldn't strictly *need* it even then).
Honestly I think you're best off NOT putting on an inner wall... it would not provide any meaningful insulation value and will just accumulate dust, shavings, feathers, mold and mice.
It for sure *is* worth priming and painting the whole inside of the coop, though. Not only will it be brighter and pleasanter, you would not believe where poo can get to, and it is far easier to remove from a painted surface than from bare wood.
If you wanted to put down a vinyl floor there should be no conflict with bare studs, b/c the vinyl would just run up to the sill 2x4 of the wall, you would not be cutting around studs unless you've built things weirdly (and if you *did* build things weirdly, filling in with 2x4s would put you back to having an even edge to run your vinyl against).
JMHO, have fun,
Pat