"The cat purred." is a sentence because it has a subject, which is the cat and a verb which is purred. A sentence fragment has either a missing subject, a missing verb, or both.
The complete subject of this sentence is "The cat." Everything from "purred" on is part of the predicate. The cat did what? Purred. How? With pleasure. At what? At the visitor's skill.
Yes because it is an action.
The old cat quietly purred near her in the garden.
The race car's motor purred like a kitten. The kitten purred when it nestled up against her mother.
Yes, I've have had a cat that purred while she was pregnant.
A complete sentence always has both a subject and a verb. If either the subject or the verb is missing, then it is a fragment. For example, "The cat sits on the bed" is a complete sentence because it has both a subject (the cat) and a verb (sits). "The cat" by itself is a fragment because it doesn't have a verb.
Do you work? is a complete sentence; it is not a sentence fragment.
fragment
"She wave." is fragment but "She waves" is a sentence.
Fragment.
sentence fragment
It is a sentence fragment