Some of the intermediates from the common ancestor with chimps and modern humans are:
Australopithecus africanus
Homo habilus
Homo erectus
Homo ergaser
These are not only intermediate in form, but also intermediate in time. Not just similar in appearance, but also changing in form gradually from one to another through time.
Once again, we don't *expect* a single intermediate, but a sequence of such. And that is what we actually find.
What happened in the genes? Well, changes in control genes for growth of the skull and jaw, mostly. Some changes in the control genes for growth of legs. Some changes in the control genes for arms and hands. A merging of two chromosomes into one at some point. Loss of hair.
There really aren't that many *fundamental* differences between chimps and humans biologically. We started having larger societies, which made us more social (or, maybe the cause was the reverse). That lead to language use, which selected for structures in the neck to make speaking easier.
No, Neanderthals, by the latest information, are a subspecies of Homo sapiens and not an ancestor.