OK. I'll try to answer the questions. Might not hit all of them, but I also have some news to break.
Army, Navy and Air Force have positions for podiatrists. None are helping while in school (no HPSP, etc.) The Army is the only one that has a residency program, which is a PM&S-36 co-sponsored by Eisenhower Army medical Center and Womack Army Medical Center. (Ft Gordon GA for the 1st year, then Ft Bragg NC for the second 2.) I'm not sure how the Navy program works, but you can be in the Navy (paid) while in a civilian residency. For Air Force, need to be out of residency first.
The Navy has some student loan repayment money available, but I am not sure of the details. The Air Force does not.
Until now, the Army did not either. This year, they decided to include Podiatrists in the Health Professions Loan Repayment Program. As of Friday, 7 were approved so some student loan help is available. It provides $32000 per year for up to 3 years with an additional year service obligation for each year. It is counted as regular pay, so 28% is deducted for taxes which means it really only puts about $20K/year towards reducing student loans. You also need to be fully qualified and licensed which excludes the residents from applying. I also see no guarantee that it will continue, but might be able to expect it to. It's been a hot button topic for quite a while. It's not as good as HPSP, but it is certainly better than nothing. (And it works out OK for me.)
If you were to do the Army residency and then apply for the loan repayment you could be looking at quite some time. The residency is 3 years with a 4 year service obligation to follow and then up to 3 additional years with loan repayment service obligation for a total of 10 years of service. For people with prior active duty time, this could put you close to retirement eligibility. Prior active duty service counts towards pay and retirement. Prior reserve service will count towards pay, but not towards retirement.
Military service has its pros and cons, like everything else. It is certainly a viable option to consider. If you search on my old posts you will probably find a bit more info.
Hope this helps.