Can I just leave a spawner trap that uses fall damage alone for about half an hour, or will the mobs recover their health, rendering the fall damage trap useless?
Mobs do not recover health. Refer to this link. However, if it is a fall damage trap, they should ideally be dying when they hit the bottom anyway so them recovering health should be irrellevant. If setup right they die on impact, and you have the ability to get in there and pickup their drops.