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No, it's an adverb. It is describing how you are doing something, not what you are doing. You don't generously; you generously do it.
yes it is a verb
No it is an adverb
generously
adv : in a generous manner; "he gave generously to several
charities"
Generously means to do or say something generously. Here's an example:
"Diane van Deren has used her gift generously, helping not only herself but many others along the way."
No, generously is an adverb, a word that describes a verb or an adjective. Example:
He generously gave his time and him money to the cause.
Generously is an adverb. The adjective form is just generous.