Wow there is a lot floating around here! Let me get things sorted if I can.

1) Neanderthals were a relative of modern man and did interbreed with our ancestors (This has been confirmed via DNA testing only come out in the last year or so.)

2) Neanderthals had a different diet than ancient homo sapiens. Their diet consisted of about 80% meat and 20% forage while ancient homo sapiens ate a diet that was more around 60% forage and 40% meat. I cannot find specific citation for this at the moment. It was a widely believed idea for a long time that Neanderthals ate only meat. While it is generally believed that they consumed far more meat than homo sapiens it is now known that they also foraged, exact percentage of what they ate likely varied with where they lived.

3) Neanderthals had high rates of arthritis because they lived very hard lives. Our homo sapien ancestors tended to take lower risk hunting methods such as the long range atlatl spear while Neanderthals used shorter range spears and were often injured hunting. One palentologist likened the Neanderthals average list of injuries to that of bull riders.

4) Now as to the paleo diet the basic idea is that hunter gatherers had a more varied and healthy diet than their agricultural offspring. There are some major flaws in this reasoning as most evidence shows that there was a gradual transition from pure hunter/gatherers to a predominantly agricultural system. In fact even the agriculture model never abandoned hunting and gathering. Those practices did not decline until people moved into cities and were cut off from woodlands and other forageable areas which is very recent. The core however of paleo diet is a good eating method BUT its most important to understand your own nutritional needs. Dairy for example. Some people have an ancestrial background that makes dairy more digestible while some have a background that makes it very difficult to digest. Furthermore on dairy there is the pasturized/raw milk debate.

Cereals are another tricky area. Generally its not cereals it is gluten cereals that are the problem and that can be broken down further between the digestability of whole grains and non whole grain cereals. A cereal is really just a seed as is a nut or any other plant seeds. Its more how they are used that makes people define them as a cereal.

My own hierarchy of food in what to eat the most of to what to eat the least of is;
Green Veggies (brassicas, artichoke, nettle, misc greens like good king henry & purslane)
Other Veggies/ Root & Tubers (squash, tomatoes, rhubarb, carrots, potatoes, rutabaga)
Meat, Eggs (chicken, turkey, bison, fish & beef)
Fruit/Berries (apples, citrus, plums, cherries, apricots, etc, wild fruits & berries especially)
Cereals/Dairy (quinoa, amaranth, chia & buckwheat [all these also act as green veggies])
Misc Food stuffs (oils, nuts, herbs, mushrooms, etc.)
Junk Food. (The crap we all eat but should not)