Walking up/down a level road/street

Simple answer – yes, you can say such things, even if the street is not sloped. The dictionary supports such usage:

down (prep.) at a point further along the course of (something) : he lived down the street. [NOAD]

down (prep.) at a point somewhere on a road, path, etc. in a direction away from you : they live somewhere down Park Avenue. [Macmillan]

up (prep.) at a further position on : a shop up the road [Collins]

up (prep.) further along (in any direction) [Wiktionary]

To keep it simple:

There is no rule regarding road vs. street. The only case I can think of where it may be important is perhaps when referring to a dirt road or rural/back-country area, in which case perhaps "street" is less common, but certainly there is nothing wrong with it even then.

There really is no rule regarding up vs. down the road. You can head "up the street" even if you're going downhill, and "down the road" when there's actually an incline.

It's a very casual phrase, and no one will call you out on it, because no one takes the time to stop and think "Wait, is there a hill here? Am I going up or down the hill? Are the numbers increasing or decreasing? Which way is up, dang it?!"

TL;DR Any of the four combinations works.

I don't think there's a specific rule involved here.

It is perfectly fine to use up/down the street/road if the street is flat. Which of the two one would use is more debatable, and somewhat depends on each particular person's point of view.

For example, one way, as you've mentioned, is to chose by house numbering: the way in which the numbers are ascending can be considered "up the street", and the way in which they are descending can be considered "down the street".

Or maybe you can consider the direction in which the street is more... erm... fashionable to be "up", and the other one to go "down".

I would say that in general, if used unrelated to the elevation, these phrases can mean the same, and can be used depending on your own preferences. Also note that "down the street" will always be understood, which I believe is the reason for it's popularity comparing to "up the street".