Not talking about followers, but those other NPCs like some named villager. If they get injured do they heal after a while, or do they stay injured forever?

Hard to tell since can't see their health bar.

They regenerate their health after a while.

They regenerate, but it's normally pretty slow.

Oh thanks, wasn't sure. What cause them to regenerate? When they sleep?

They regenerate health like your character does. It might just take a bit longer for them.