When used in respect of spatial relationships, the core sense of over is above (vertical displacement). But it also has strong associations with beyond (horizontal displacement) and across (calling attention to the intervening space or pathway which must be crossed/traversed to get from here to over there).
Looking at OP's specific examples, if you say a building is over the road/street that will almost always mean on the other side (i.e. - opposite the side the speaker is on, or the "current" side being spoken of). If the concept of crossing the street isn't relevant, the speaker would use up or down, not over.
Idiomatically, you wouldn't normally walk over the street (you either cross it, or walk along it).
But "Come over here" doesn't necessarily imply anything more than crossing the intervening space. You can just as naturally call this out to someone on the same side of the road, bridge, or other barrier as yourself. The actual distance doesn't need to be far either - you could reasonably say "Sit over here" to someone who's already very close to the seat you're inviting them to take.
TL;DR: Whilst over can have connotations of being above something, or crossing a "barrier", it can also be used in contexts where the only thing being traversed is a (literal or figurative) distance. The usage "dates back over centuries".