Why is there no definite article before "closest" in the question "Who are you closest to in your family?" My only assumption is that "to be close to someone" is a set phrase and it is used without an article.
Example sentence with article and superlative:
Which hospital is the closest?
Note that in this example there is no prepositional phrases, i.e., "~ to xxx". Were we to introduce such a phrase, we would remove the article:
Which hospital is closest to this house?
Which hospital is the closest [hospital] to this house?
You can see that it's overkill to put both in. It is not necessarily grammatically wrong to use both if you structure it as such, but it might be considered semantically inelegant.
The "to" in your original example, note, is part of the verb phrase "to be close to", as you correctly pointed out. There is a separate prepositional phrase "in your family":
Who are you closest to in your family?
It appears that using that verb phrase with an article renders it ungrammatical, maybe. Certainly somewhat awkward:
Who are you the closest to in your family?
However, you can just use close intransitively and omit the reflexive prepositional phrase (e.g., We are close [to each other is implied]). In that case, you could use a definite article with the superlative:
With whom are you the closest(, in your family)?
Only nouns take articles. "Closest" is an adjective, not a noun. In the sentence "Who are you closest to in your family?", "closest" modifies "you", and "you" needs no article.
Carly gives the example sentence "Which hospital is the closest?" This can be understood as being short for "Which hospital is the hospital that is closest?" In English, there are situation where a noun being modified by an adjective can be dropped, and the adjective can stand in for the noun, and that is happening here: the noun "hospital" in "closest hospital" is being dropped, leaving the adjective "closest" to act like a noun, and it is that capacity that it receives the article.