I'm a little undecided on the "Foothill Freeway" example, but the ambiguity seems to hinge more on dropping the "freeway/blvd" portions than the definite article.
But for Interstates, having a "the" seems a bit silly and it doesn't resolve any ambiguity that I can tell. I wouldn't say "Go north on the 15." I have multiple problems with that. There's the already discussed point of it being "Interstate 15" and not "The Interstate 15," of course. But, I'm not digging the "go north" part either. That is, Interstates are labeled with directions already. So I would say "take 15 North." "North" being part of the proper noun. I-15N spends a lot of time going northeast, and I imagine occasionally turns southward briefly as topography demands.
If you're going to Vegas from San Diego, you're taking I-15N to Las Vegas. Or you could say you're going northeast on I-15. Or northeast on I-15N for extra clarity.
As for up/down. I guess I generally think of it as north and south, but I don't think I take it as a direction. If someone said "go down I-15N" for example, it wouldn't bother me. I never say 'go over to' because that's going crazy overboard with the syllables. Sometimes up/down is used to refer to actual elevation changes. I know it was like that when I lived on Lake Superior. Up was away from the lake. Down was toward the lake. I guess to some extent it's like that here. People say "I'm going down to the beach," and not "over to the beach" even though it's straight to the west.