You are talking about a specific road so I don't think it's not possible to leave out the definite article "a".

To see how strange it is without it, try taking "the" out of "to whom belongs the road?". In English the define article is "imbedded" in "whose".

That would be "who is the owner of the road." Similar, but not what the question wanted unfortunately.

"Cuja" is used as a relative pronoun, that is, when you have the main sentence and a subordinated one. The translation for "whose" in a question is "de quem".