What is the grammar of the above sentence? Is "to a dollar" a prepositional phrase? Does it function as an adverbial or adjectival phrase?
It is the same as "There are 5 fingers on my hand". "There" is dummy subject, the verb is the 3rd person plural form of "be". "100 cents" is the complement, and "to a dollar" is a prepositional phrase, which is being used adverbially. Compare "There are 100 cents in my pocket", here "in my pocket" is clearly an adverbial, modifying "are", indicating where the 100 cents are. Comparably, "to a dollar" indicates what the 100 cents make. It's adverbial, but it's not the clearest example.
The word "to" is used idiomatically. The word "in" may also be used, "to a dollar" seems more typical of US dialects; "in a dollar" might be more common in Britain.
It means a dollar is made up of 100 cents.