I have a question about the use of contractions with a proper name. For example, if I wanted to write "Mark's no longer a member", is that correct? Or do I have to write it as "Mark is no longer a member"?

What is the rule that defines the use of this?

I eagerly await your reply. Thank you. Emotion: smile
Hi, and welcome to English Forums.

This is fine to say, and fine to write in many contexts. Some people believe that contractions belong only in the most informal writing situations. You need to decide whether the contraction is too informal for your situation.
Whether you use the contraction or not depends on how formal you want to be. The contraction is informal.