Is it permissible to run romex without a conduit on the ceiling of your garage
When the garage is enclosed with sheet rock and the romex will be on the
Exposed side of the sheet rock? If a conduit is required will it have to be
Metallic?

You could consider a garage as a damp location in which exposed NM cable is not permitted.
Conduit can be metal or PVC, your choice.

Gordon,

I noticed your other post wit the same question and asked to have it removed. Please refrain from asking the same question in different posts.

If you need more assistance on this thread, just append to this thread.

Don