I hope I can still graduate on time but I'll have to check with my school. Let's say, hypothetically, that my school allows me to graduate on time but I have to do a rotation after graduation, do you think a residency program in primary care would care? I don't know what the timing is like for the gap between graduation and the start of residency but do you think I'd have time to do a 1 months rotation after residency and still start the residency on time? When I apply for residency will the program even know that I am behind my school's schedule?
First, definitely agree with everyone that you need to check with your school to see if you can postpone your test, and whether it will have any impact on your progression towards graduation.
Whether someone will noticed that you took COMLEX in September instead of June - probably not - programs get inundated with applications (and each application packet is quite thick, even in pdf format) they'll probably not noticed the date. Now your MSPE (formerly known as dean's letter) will have details about your preclinical years, your clinical years, how you compare to your peers, and usually a summary page which may or may not include issues that you are dealing with right now. I've seen some medical schools even include summaries of the applicant's undergraduate transcripts in their MSPE (these were MD schools if I recall, and not all of them, just a minority of them). A question that we (the anonymous internet) cannot answer is whether this issue is a one-off, or would it be re-occurring issue (ie comes up again during residency). That may pose a challenge to residencies (but again, since I don't know the details, and there is no need for you to tell us the details - it's all speculation). Talking with your school administration as well as your faculty advisor should be encouraged.
As for whether you can graduate, finish your rotation, then start residency - it depends. You may participate in graduation ceremony, but you won't officially "graduated" and get your degree until you fulfill all academic requirements. Some schools only confer academic degrees at certain time of the year and if you are off-schedule, you may not officially graduate until the next "graduation date"
If you haven't officially graduated, you cannot start residency. If you haven't officially graduated, you cannot get a medical training license. It is not uncommon for residents to start off-cycle, a few months behind. But keep in mind that unlike academia, as a resident, you are now part of a bigger picture - and schedules (wards, calls, clinics) will have to be made, even if you are absent. So someone will have to fill that gap until you can start. Most residencies are willing to work with you (if the start date is reasonable). If your start date is too far out, then they can consider it a match violation (since you can't start on time) and can terminate your contact.
Lots of variable - lots of unknown. You're asking for specific answers to vague scenarios and situations. All we can give are vague possible answers. Best advice is to talk with your school.