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"More shyly" is the comparative of "shyly". There is no single word form as with lowlier.
Sure! "She smiled shyly when he complimented her on her artwork."
The adverb of shy is shyly.
No, "shyly" is not a verb. It is an adverb that describes how someone is behaving or acting.
"Shyly" is an adverb. It is used to describe how someone is behaving in a timid or bashful manner.