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Yes they are correct. When the writer wants to stress the individual members of the group, it is possible to treat the collective noun as a plural. The noun is treated as a plural and the verb form is plural.
Up could possibly be classified as an adverb, but it is often used as a preposition (up the hill) and like a number of other prepositions it is often paired with a verb in idiomatic pairs like blow up so it may be better to treat it as a special kind of preposition.
Yes, this verb is modified by an adverb. You might also say that the adverb modifies the verb chain containing this verb, or even the clause or sentence containing it, but the main point is that this verb's meaning is affected by the adverb.
To is not a preposition when it is combined with an infinitive as in to pick. It is a matter of debate how it should be classified, but although it used to be a preposition it certainly is not one now.
Until can be used as a preposition (until Christmas) but here it is a subordinating conjunction introducing a subordinate clause until one day they saw an egg hatching ....... finally I came to a story about a fig-tree. This fig-tree grew on a green lawn between the house of a Jewish man and a convent, and the Jewish man and a beautiful dark nun kept meeting at the tree to pick the ripe figs, until one day they saw an egg hatching in a bird's nest on a branch of the tree ...