Should I mention that I am accepted into an MPH program through a medical school which I will do during the gap year as part of "continuing education"? I have also taken post-bacc courses which I briefly mention as what I did to raise my academic credentials after graduating due to low GPA
I think you need to be careful in this situation. It is a catch 22. I say this because the med school that you are currently doing your MPH, or prospective dental school, might frown upon the fact that you just started this program. This is considered a professional school, and within health care, and might seem to some as competition. Especially, if you are not going to complete the degree. Plus, your current school will not like the fact that you are using them when they could give that spot to someone who is going to complete the program and benefit the local community.
On the other hand, prospective dental schools, will love the fact that you are doing this program. None the less, all of your academic transcripts must be sent to the dental school in your application. So they will know about it and all of your "post-bacc" studies. I think what you need to do is be able to explain why public health and how it pertains to dentistry. If this is what you truly want to do, then complete the degree, then apply to dental school. Then say you want to do public health dentistry and they will eat that up.
But, that is my two cents, and ultimately it is your decision and only you know best what suits your life, education, and future career.