Is this sentence a fragment Talking on the phone is her favorite pastime?

Answer 1

No, because it has a subject and a verb. "Talking" is the subject (it's a gerund: a verb used as a noun. They usually end in -ing, like swimming or running), and "is" is the verb.

Answer 2

Yes, the given sentence is a fragment because it lacks a main verb. A complete sentence must have a subject and a predicate containing a verb.