The potion and wand version still use the spell rules, so yes, none of them automatically work. That's a good pickup, actually - I hadn't noticed the change from 3.5!
A Periapt of health just straight up makes you immune to disease, no roll required.
Heal.(the spell)
Can you pass a periapt of health from person to person to heal them like an inexhaustible potion?
No. It only makes you immune to disease, it won't remove it. At least in the real world, you can be immune to a disease but still a carrier.
Should you ever become un-immune (by say, taking off the periapt) the effects of the disease would apply again.
I don't think there's a "carrier" mechanism in PF though...
Not really, no. But something similar does exist, like how ghouls can spread ghoul fever, but are not affected by it themselves.
Though there is the Plague Bringer antipaladin class feature.
Plague Bringer (Ex): At 3rd level, the powers of darkness make an antipaladin a beacon of corruption and disease. An antipaladin does not take any damage or take any penalty from diseases. He can still contract diseases and spread them to others, but he is otherwise immune to their effects.
Specifically immune to disease, but can still contract spread their effects. So there is some precedent for being immune to disease, but still having the disease in you.
At best each day when you would make your disease saving throw, the periapt would make it an auto-success, curing you in 1-3 days (unless it couldn't be cured by saves).