My house is down the street.
Does the adverb down modify is, or street?
Down here is not an adverb but a preposition, heading the prepositional phrase down the street.
The PP as a whole acts as — well, now it gets tricky. The grammar I grew up with would have called it an adverb of location modifying is:
But the older I get the less I like that. I'm inclined to say that the PP acts as an adjective of location modifying (actually, “predicated of”) house:
Which all goes to show how much more flexible the language is than the categories we use to describe it.
But in real life it's whatever your English teacher says it is.
Neither. Down in this situation is a preposition, not an adverb. The complete phrase is "down the street". You can tell it doesn't modify either of the two words because it can't be separated from them:
Neither makes sense.