If you follow the rule that an independent statement has to have a verb, you cannot consider "loitering" or "winning" to be verbs. They are both nouns that describe an activity in a way that depends on the main clause for meaning.
Consider an alternative, "after he loitered for ten minutes, the boy walked away." This structure inserts a dependency by using "after" to arrange the two acts. Remove "after" and you have two independent statements. Neither "after loitering for ten minutes" nor "loitering for ten minutes" is an independent statement. "Walked" is therefore the only verb in sentence one.
Applying the same rule to sentence two, "has been boasting" is the only verb.
Sentence three is a great point for argumentation, depending on how you like to diagram the structure. You could argue that "having settled" is an idiomatic version of the past perfect tense with a progressive mutation, in which case it is a verb. Or you could argue that it is an idiomatic gerund with a progressive mutation, in which case this sentence is the same type as the first two. I favor the latter.