Asking the important questions

Or if they were knighted would they be greeted as Dr Mr Sir President

Sir Gen Dr. Mr. President OGBYN B.S. M.S. M.A Ph.D. Sc.d.

In answer to the question you use the higher title in a social situation. If you met a general/ doctor Fred, you'd call him General Fred. If you were corresponding with him about his scientific research though, you'd call him Dr. Fred. In correspondence you use the title prevalent to the subject matter. When in doubt use the higher title. (So it would just be President) btw you never combine titles, one at a time

Well the normal greeting for the president is "Mr.president" so I assume it would be "Dr.president".

Congress decreed in 1792-3 that the president shall be known as "Mr. President". So, to answer your Q, no.

That is Mr. Dr. Professor Patrick to you my good fellow!

Moot point.
Large Handed and Penised Overlord Trump will be deporting all doctors.