No, it is not a compound predicate. "While" is a subordinating conjunction there. In the first sentence "and" is a coordinating conjunction.
I know that the following sentence has a compound predicate:
Susan held and petted her cat.
My question is, does this next sentence have a compound predicate?
Susan held her cat while petting it.
And, if not, can you explain why it doesn't?
"Susan held and petted her cat."
Does the above sentence really have a compound predicate? If so, it is different from the predicate in 'Susan held her cat and chased the dog away.'