I think it is fine to paint the interior of the coop. -have known a number of folks that have painted their coop interiors. (Obviously, you wouldn't want chickens closed up in the coop with fresh paint.....but after it dries and airs a bit, it should be fine.)
I used a mildew-resistant primer and then an oil-based semi-gloss (found a great deal on the clearance rack!) and had them tint it sky blue for me. The coop was nearly done before the chicks arrived so it had weeks to air out but it didn't need more than a few days. And it's still pretty more than a year later! I also like that it's easier to clean than unpainted wood, it's sealed against moisture, and the light reflects making it brighter inside.
Check the mistint pile, Freecycle, Restore, etc. to get high-quality paint on a budget. You don't want to invest a lot of time in painting only to have the paint not last.
I used Ooops paint from the paint store, in semi-gloss. It cost me $5.00 and it was nice it was just a soft grey and not a red or something. I do the deep litter method and just last week cleaned out the litter. I do it every 5-6 months. And the coop was clean and dry easy to just scoop out the litter with a dust pan.