Re: our neighbor down the street [=our neighbor who lives farther down on our street]
That looks like a matter of a many, many blocks. I wouldn't say "My neighbor down the street" for that.
Lets say you have the main road. Then you have your neighborhood street coming off the main road. The families in order are A, B, C, D, and E.
B, C, D, and E live down the street from A.
C, D, and E live down the street from B.
D and E live down the street from C.
If C and D are talking about E, they may say "You know, the Es. They live down the street." If they are talking about A, they may say "The As -- you know, the folks up the street."
There is often no clear "up" or "down": it can go either way. C might say that A and B live down the street and E and F are up the street.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.