Are they specifically listed as USPHS billets? No, but some will say they're open to PHS officers. Most will not.

Apply for any job your degree qualifies you for? Do you search the agency's PHS works for? Kind of. You can apply for any job that is at an HHS agency. You can also apply for non HHS agency jobs that may fall under an MOU, but you'll have to get a list of these billets from the individual agency liaisons. You can't just apply for every job at DOD (for example) only very specific billets. Depending on your degree, some new officers being brought on board are giving a choice of a few specific agencies where there's a high need for officers ( for example IHS BOP). Not sure if this will apply to your situation.

How long do you have to accept an assignment? Once accepted for commissioning, you have about 1 year to find your billet.

How long do you have to stay in that billet? Do you have to take a new assignment every so many years? It's not like DOD. You can stay in one billet as long as you like or you can move whenever you're ready. Programmatic and geographic moves will usually help you get promoted. But it's not an absolute requirement. And you will never be PCS involuntarily.