Question

(1) Glen’s horse, Tipper, grazes lazily in his grassy.In sentence (1), what part of speech is "lazily"?

Answer 1

In that sentence ... and most likely wherever it's used ... "lazily" is an adverb,
modifying the verb "grazes".

Here's a tip:  Whenever you see a word that ends in "...ly", it's usually an adverb.