Yes, "may have" is a modal verb phrase indicating possibility or permission.
yes. A passive verb phrase.
Yes it's a verb phrase.
yes
will take is a verb phrase
Yes, a verb phrase can be split by an adverb or adverbial phrase. For example, in the sentence "I will quickly finish my homework," the adverb "quickly" splits the verb phrase "will finish."
Yes. "Will eat" is a verb phrase, qualifying it as a verb in the future tense.
"Had become" is a verb phrase where "had" is an auxiliary (helping) verb and "become" is the main verb. The auxiliary verb "had" indicates the past perfect tense.
Yes
yes
Yes, "picked up" is a verb phrase formed by the verb "picked" and the adverb "up."
Yes, "studying" is a verb phrase consisting of the main verb "study" and the present participle "ing." It functions as the action in a sentence.