I don't follow anthropology real closely, so I can't tell you off the top of my head whether there are or are not fossil records of "groups of humans migrating to China from Africa". There may well be, for all I know.
But even if there aren't, we can confidently conclude that the ancient east Asians migrated to their current location from Africa, due to the following observations:
1. There are no human or even proto-human/pre-human fossils in China (or anywhere else aside from Africa) prior to about a million years ago.
2. Human, proto-human, and pre-human fossils are found in Africa, and Africa alone, prior to that time, and stretching back millions of years earlier.
So obviously, humans first arose in Africa, and then migrated outwards from there into China (and most of the rest of the planet).
Also, for what it's worth, despite their early arrival, East Asians didn't begin to acquire agriculture or tools beyond the stone-age level until around 7500BC:
I'm leaving out a *ton* of detail and corroborating evidence, of course. But that's the Cliff Notes version.
Here's a map of the major human migrations:
The previous two figures are from the book, "Guns, Germs, and Steel: The Fates of Human Societies", by Jared Diamond. It's a really fascinating compilation of what's currently known about human pre-history, including migrations, changes in languages, and the origination(s) and rise of tool-using, agriculture, animal domestication, modern civilization, writing, and so on. And all this is background for the main focus book, which is the question, "why did some peoples advance more quickly than others"? For example, why were Europeans the ones to cross the ocean and conquer the native Americans, rather than vice versa?
You might want to read chapter 16, "How China Became Chinese". It reconstructs how China became the monolithic cultural entity it is today, despite starting out as countless small scattered groups of culturally, linguistically, and genetically diverse peoples.
When/how did the Chinese get there? Why are they SO different (physically) from the other races? (I use "race" to simplify the arguement) We can see that Africans look like Semites look like Indians.
Again, read the book. The author also points out that although most Asians tend to just "look Asian" to Caucasians, there is actually a huge diversity among Asians:
"We take this seeming unity of China so much for granted that we forget how astonishing it is. One reason we should not have expected such unity is genetic. While a coarse racial classification of world peoples lumps all Chinese people as so-called Mongoloids, that category conceals much more variation than the differences between Swedes, Italians, and Irish within Europe. In particular, North and South Chinese are genetically and physically rather different: North Chinese are most similar to Tibetans and Nepalese, while South Chinese are similar to Vietnamese and Filipinos. My North and South Chinese friends can often distinguish each other at a glance by physical appearance: The North Chinese tend to be taller, heavier, paler, with more pointed noses, and with smaller eyes that appear more "slanted" (because of what is termed their epicanthic fold)."-- Ibid, p. 323
Where did this change in skin tone and bridged noses come in?
Obviously, the ancestors of the modern Asians genetically diversified from the groups of humans who migrated elsewhere.
Why does it effect them universally?
See above -- there actually is a lot of diversity even among "Chinese". But whatever "universal" traits they may have can be explained by either (or both) of the following: 1) those "universal" traits arose in the "founder population" before subsequent radiation across the rest of Asia, and/or 2) they arose in a subpopulation after such radiation, but spread back through the whole population via subsequent interbreeding.
Why are caucasians the only ones capable of getting red hair genetically?
Actually, it's not clear that they are the "only ones". When Pizarro entered the Incan empire in 1532, he noted that although most of the Incans were dark-skinned and short, a substantial number had skin lighter than that of the Spaniards -- and red beards. And when Captain Roggeveen landed on Easter Island in 1722, many of the inhabitants had white skin, with blue eyes, and red hair.
Red hair results from a mutation in the MC1R gene, and DNA analysis indicates that the European variant dates back at around 75,000-500,000 years. So to answer your question, the reason that most red-haired peoples are European is because the most common mutation of the MC1R gene occurred within Europe after the initial "out of Africa" migration.
See:
The Melanocortin 1 Receptor (MC1R): More Than Just Red HairMelanocortin 1 receptor: What’s red got to do with it? [You've got to love a paper that has a section heading like, "The genetics of red hair and pale skin: First, the dull biochemistry" -- Ich.]
If we all came from africa, wouldn't red hair be possible in all the races? Surely there are mild climates in Africa and India that would allow for lighter colorign of people.
See the above papers. And just because a particular mutation would be "possible" in all races, doesn't mean it has to happen everywhere.