I've seen two kinds of mouth opening.
The intermittent kind, like hey have something stuck in their throat that they can't get down. Every now and then they each up and look almost like they're yawning. Can you look down his throat?
The other kind is that mouth opening and closing over and over in a row, like gasping, and this is what chicks do just before they die. I see this in extremely young chicks most often, like the ones I have pulled out of their shells them not having hatched on their own. If they are going to die they do that open mouthed gasping thing, then expire.
Closed eyes and a sleepy look is not sleepy. It's pain. Some birds can be hunched and ruffled and sick, but still alert and seeing their environment. When they get that closing the eyes thing, they are hurting.
Hot birds also pant, mouths open, breathing like a hot dog. Not a hot dog on a bun, but a hot dog who's been chasing a stick. Didn't want to confuse you. Good luck.