Did humans evolve from neanderthals?

I'm a bit confused on how this occurred...did a neanderthal have a weird looking child, with mutations that made it a human? Could you possibly explain how they evolved? Next, humans are thought to have originated in africa, but there isn't any fossil evidence (according to sources I've read) of neanderthals living in africa. How is this explained? Lastly, it is documented that humans and neanderthals interbred. If the divergence occurred hundreds of thousands of years ago, why didn't they interbreed back then; evidence for gene flow only is only documented within the past 50-80,000 years.