Usually it is just primed and the tiny nails used should leave very tiny holes and in my experience best just filled with drywall compound and of course you need to paint all of the baseboard with 2 coats of paint IMO

Ours came primed and prepainted. I thought that was going to save me time. Seems like I should have just gotten primed.

Can you find a caulk or putty that's close enough in color to just putty the holes?

Its better to just put 2 coats of paint on it, since there are corners and miters, etc in trim. You want it to look really nice and finished when you are done with it. I prime trim, and prepaint it one coat, and then fill the nail holes, light primer on it, and then brush a coat of paint over the nail holes, and finally, do a whole second coat.

if you can still see the holes then you didn't fill them high enough or sand them perfectly smooth w/the surface. I've done a million of these and they take a little finesse.

As for the sheen, what sheen is the paint for walls, and the baseboards? is one glossy and one eggshell? did they give you touch up paint for the walls? if so, it will be flatter than the paint used for trim. are you sure you got the right touch up paint for trim? spraying or brushing wouldn't make a diff in sheen.