They've done well for thousands of years by hanging around us humans.
We don't feed them on purpose but somehow the little things find a way.
Their bodies are made of cartlidge which allows them to squeeze through tiny spaces - like the small entrance of an entranace reducer.
I haven't had mice in a hive... I hope that's something I never experience.
If the bees did sting and kill a mouse inside the hive the carcass would lie there and rot. It's too heavy for them to remove. So instead the bees will leave the body in place and cover it entirely with propolis, thereby sealing it away from the clean hive.
Last fall, the lid on one of my plastic bins didn't fit down tightly. I had jammed too much bee equipment in there - rim spacers, extra inner covers, queen excluders, etc., and a couple green drone comb frames.
And that was a mistake.
No less than five mice moved in. I discovered it in the fall when doing a pre winter clean up of the bee yard.
They'd made a grass nest and of course the smell of urine was pretty strong... one more thing to clean up before winter.
Then I removed extra equipment so the lid fit down snug. Lesson learned.