Is it common for the plaintiff to move for JMOL after the defense has rested in a medical malpractice case?

You can make any valid motion you want during a trial. However, most judges are leery of appellate courts if they are appealed, so they are usually reluctant to make this type of ruling. They would rather err of the side of letting it go to the jury. These types of motions are very fact specific so it is impossible to give you a yes/no answer without being privy to the intricacies of your case.

It is common for both sides to move for directed verdict. It is often denied but preserves appellate rights

Yes, one very commonly might do so (both Plaintiff and Defendant(s)), will typically move for directed verdict at the close of first the Plaintiff's case, and then again, at the close of the Defendant(s)'s case(s); however, it is typically done so in a fairly perfunctory manner and often to merely preserve appellate issues. If one thought that this type of Motion was likely to actually result in a Directed Verdict, the issue would have been vigorously raised via a motion for summary judgment at the close of discovery during the pendency of the action. This is a very rare scenario in which one might expect to prevail. Typically an action in which the doctrine of res ipsa loquitor is applicable is the sort in which one might expect reasonably to prevail. A good example is a surgical instrument left in the body, for example, and further proof that the instrument count was never conducted, or at least not in conformity with protocol and the more importantly the community standard of care. And yes, one can move only as to liability, of course. Typically when the motion is made that is the issue to which it is primarily if not completely directed.