Any kind of paint should be fine on wood,
as long as it's allowed to fully cure at the right temperature
before the birds are exposed to it.
Why do you feel the need to paint the inside of coop...need more light...add some windows?
Not sure there any paints that will stick to that kind of plastic long term.
Nothing uglier than paint peeling and/or scratched off plastic...BTDT.
Are the nesting boxes plastic too?