Well, there is a C flat, it's just also called B. And vice versa for B sharp, that's just C. There is a half step between B and C, it's B to C. All the sharp or flat symbols do is signify you should play the half step up or down, respectively, from the decorated note.

The same is true between E and F. There's a bunch of complicated reasons that the black keys have gaps between them, ranging from the physics of sound, to Western music theory principles, to ergonomics. I'm by no means an expert on any of this, I just think it's interesting, and a fun topic to look stuff up about.