You are correct. They have found Neanderthal in Israel (near Carmel) that dated to about 92,000 years, and also Homo Sapiens Sapiens that dated about 102,000 years - we were contemporaneous with them (NOT descended from them!) Apparently, they were Hunter Gatherers and limited intelligently, while the Homo Sapiens Sapiens were farmers and more family oriented and had a better language and intelligence. Why Neanderthal died out, and we did not is a mystery to Anthropologists...
Yes, those caves in the Levant are especially intriguing, because it appears that modern humans moved in, displacing Neanderthal, then moved back out, whereupon Neanderthal moved back in, and then modern humans returned with Neanderthal vanishing for good. The best conjecture I've seen is that climate changes accounted for the migrations of the hominid populations.
Lots of possibilities. For example, some Japanese researchers did a gait simulation from some skeletons. It looks like Cro-mags would have been better runners, which is a BIG advantage in conflict -- if you can attack faster than your opponent can retreat or maneuver, and run away faster than he can attack, then you tend to win in conflicts