I suggest not installing one at all, unless you live in a Victorian...

I agree with acm. Could we see the room and the furniture?

Agree with acm. Unless your house is super traditional/period, I wouldn't do it. We removed all of the chair rail from our house quite a while back.

Install it. Then paint once installed for the simple matter of filling imperfections, nail holes, and anything unsightly. It also looks much better cleanly caulked to the walls with the caulking painted in the wall colour cleanly, leaving a nice sharp line. It could be primed before being installed. I never trust the factory primers (in the case of coming pre-primed) and use a high performing primer. But top coats do once installed. It does not necessarily have to be sprayed once in place. There are some pretty slick looking modified oils (now called waterborne emulsions) that can produce very nice glass like finishes if applied right.

I would much rather have NO chair rail, if it were me. But install it before painting. You can prime it before, but install and then caulk, then paint.