1. non accredited means non-ACGME, so you cannot sit for the boards (ABMS specialties)
2. I believe (others can correct me if I am wrong) that combined programs will enable you to sit for the respective boards, but you have to fulfill the specific requirements of the given board you are applying for (i.e. the correct number of weeks of the required rotations). Thus for med-peds, you'd have to fulfill the minimum peds and/or the minimum IM requirements to be eligible to sit for the boards. I would think that they automatically fulfill the criteria, especially as most people end up picking one over the other, but if not, this may mean that you need to use some elective time to get more time on one or the other specialty (?). This is a better question for those in a combined residency program as I am speculating on this detail....However, I've NEVER heard that combined programs are board ineligible. It would defeat the purpose of the training in the first place.
3. A non-ACGME accredited residency *could* severely affect your ability to get a fellowship. Many fellowships only want a board eligible person, and coming from a non-ACGME program, one would not meet this criteria. And fellowship PDs want you to sit for the specialty boards, which means you need to pass the initial (IM, peds, GS, etc.) boards as well. This is not as likely to be a problem with non-ACGME fellowships, but as I said, you still would not be board eligible....if you put in that much time and effort, get something in return. Patients are savvy and can look up your credentials. They know BC status is important.
4. The biggest problem with a nonACGME residency is that you can't be boarded....and most employers and hospitals are now requiring board eligibility and/or certification to maintain privileges. Some insurance payors also want BE/BC status to get you on their plans. No privileges or no insurance coverage means you are not employable.