Yeah, it'll affect everything.
For the record, I know an Army physician that graduated medical school, started residency, and ended residency late. I know no other details of that situation, but it's at least possible. One thing that I hope is obvious to you is don't keep this a secret with your residency. I don't know if you want to tell anyone before the match, especially if you consider yourself a marginal applicant. If you feel like you're a strong candidate, then full and upfront disclosure is probably the way to go. In any case, just don't show up in June and drop this bomb on everyone. I know it sounds crazy, but I wouldn't warn you if others hadn't done similarl things.
Only physicians are in the medical corps, and - obviously - you're not a physician without a degree. I don't think you'll get promoted until you've graduated. That obviously will affect your date in rank and pay.
The trickiest part will be your orders. I'm guessing you will have exhausted all ADTs on audition rotations at this point. That means, if your orders don't call you to active duty until after you have your degree, then you'll spend the months of June and July living off of savings. That is, unless you convince someone to call you to active duty while still technically a medical student/MSC officer.
Regarding orientation, I'm sure you'll be able to do a lot of the things, like hospital orientation, ACLS certification, etc., but there's a lot of administrative crap that you won't be able to do until your orders are valid.
What specialty are we talking about here? Or have you decided? Have you considered what will happen if you match a site that doesn't typically have medical students rotate through the ED? Will you have to establish a MOU between this hospital and your university?
I'm guessing you've run all possible avenues of graduating on time to ground. And is there no chance they'll let you do BOLC between internship and residency? That's not ideal either, but it's better to go TDY once you're already in the system rather than push back entry into the system. This is an ambitious plan that will take a lot of perseverence and diligence on your part. Dealing with bureaucrats isn't easy even when you fit into their little box of what to expect. You've taken that box, painted it pink, and turned it into circle.
Good luck.
