"Health care group" isn't a phrase that gets you very far. I think you are asking if there's a way for you to target anesthesia employers based on tax status, and I doubt you'll find a list. Also the info changes over time as hospitals buy practices and as practices buy each other. In your shoes I'd target a handful of regions, and start building your own list, maybe starting from individual MDs. For instance, pick a hospital, see if anes is one of the specialties by which you can search providers, and then dive deep on each provider to see what practice they're with. For instance, google "anesthesia Milwaukee" to find the anes practices there. I expect that individual practice group websites at least hint at their status, and over time you can get your own mini-MBA by learning how to tease out the info you need. Maybe others can suggest ways to search for tax status of a practice group that includes MD anesthesiologists.
Not necessarily. You'll need to be the grownup in charge of caring about the actual tax status of your actual employer at a residency program. There's no defacto nonprofitness about residency. Not all residencies take the usual Medicare funding.
You also need to be the grownup in charge of knowing whether it's likely or beneficial to stay at your residency site for fellowship. I have no idea if 1 year is the norm for anes fellowships or how competitive/regional the various subspecialties are.
Let's throw that back at you: what do you perceive as the benefits of doing forbearance vs. starting IBR? Do you anticipate not being able to handle the IB part of IBR during residency, by which I mean do you have valid reasons to not be able to afford the federal loan payments calculated Based on your Income during residency?
I totally understand the terror of massive student debt, but I suggest that PSLF eligibility may not belong on your top 5 list of decision points in planning your career specifics. As you'll see by reading the PSLF discussions here and elsewhere, the stability of the program and its friendliness for helping those at the top 25% of income in the US is not a given.
And I suggest that you're way off the reservation in trying to combine PSLF, anesthesia, private practice, and income levels over $300k. You're unicorn hunting. If you meet an actual unicorn fitting your parameters (anes MD/DO in practice, successfully maintaining PSLF status through employers that will still be non-profit in your own time frame), then maybe it's more possible than I'm describing.
Best of luck to you.