It's interesting how much Chinese people like to talk about horses. I mean, the word "quickly" in Chinese literally means "on a horse" （马上）...
Maybe it has historical roots, since horses played a big role in Central Asia?
Surely "street" should also be accepted.
Literally it should have been, but we always say 過馬路/过马路 only, be it a street, a road, an avenue or a boulevard that we are crossing. It is about the specific action, I think.
What's the reason for the first character of "road" being 马 (horse)?
马车, which is a horse-drawn carriage, was the form of transport preferred by those who could afford it and required 马路 for the horses to draw the carriage along. Others travelled on foot and thus could go on smaller paths.
Used to mean "horsepath", but the term came to refer to any road.
I think 'do not cross here' shuld be accepted as well. Like this the sentence strongly implies crossing the road, unless context suggests otherwise.
Why is 过 required here?
过 (guò) means to cross.