Yes, convince, as to persuade somebody, is an action, therefore
it is a verb.
A verb is a word that describes an action (run, walk, etc), a state of being
(exist, stand, etc) or occurrence
(happen, become, etc).
No, the word 'convinced' is the past participle, past tense of
the verb to convince. The past participle of the verb also
functions as an adjective.
The abstract noun forms of the verb to convince are conviction
and the gerund, convincing.