The meaning of adverbs

1) Yes, "at the park" and "again" are both modifiers of "played." The former is an adverbial prepositional phrase, and the latter is a simple adverb.

2) In this case, you are once more correct in your thinking. Both adverbs modify "played," though this is not a phrasing one would often use; a smoother way of saying the same thing would be "Again, I played quickly."

In this case the adverb modifies the prepositional phrase, due to its placement. Were the adverb to precede the verb, it would modify the verb.

When there are multiple eligible principals to modify, placement is paramount.

English adverbs don't modify verbs (with the possible exception of degree adverbs). One reasonably well worked out scheme characterizing English adverbs is McCawley's, where clause adverbs are distinguished by the type of constituent they modify. See chapter 19 pp. 666- of his The Syntactic Phenomena of English, some of which is available on line: SPHE.

I think again is a modifier like only and too which, also following McCawley, have a scope and a focus. The scope is a constituent establishing a basis for a contrast, and the focus is an item contrasted which is contained somewhere within the modified constituent.