Do I need to have a "May not continue at University" on my transcript to be able to apply for reinstatement? How about if my grades are relatively low, regardless of specific courses or any specific program, in addition to supporting medical reasons? What are the chances?

Can't you just email the academic adviser with specific and rare questions like that?

After all they get paid with your money.

You can only be reinstated if you have been kicked out of the school. If you haven't been kicked out, "the chances" of reinstatement are zero.
For other circumstances where the grades are low but not low enough to be kicked out, you should talk to your academic advisor.