An interesting question and one that, perhaps unexpectedly, ties in with the emergence of the infamous Proto-Indo-European language group. Quick disclaimer: there's currently heated discussion about the origins of PIE and how this ties in with the wheel (and forms of agriculture) so I'll do my best to keep some perspective. PIE, for the uninitiated, is the ancestor of such diverse languages as Hindi, English and Icelandic and represents a massive cultural shift during the Bronze Age.

The wheel first shows up in the archaeological record around 3500BCE in Poland as a diagrammatic marking on some pottery found in a Funnelbeaker settlement; a full wheel was found in a Slovenian marsh and was dated to approximately the same time. Spoked wheels are found somewhat later around 2000BC, probably by the Andronovo culture on the Eurasian steppe; wheels in the 3rd millenium BC were solid wooden disks with a simple hole in the middle for an axle.

The wheel also shows up in Sumerian and Caucasian archaeology at roughly the same time. It's possible to track the spread of the wheel across Europe and Asia by dating archaeological finds. It would seem the wheel was invented in one place - probably somewhere in what is now southern Russia - and spread outwards reaching China somewhere before 1500BC and finally gracing Britain in 500BC. The later innovation, spokes, originated in what is now Kazakhstan and expanded somewhat more violently than the previous format - spoked wheels spread with the chariot, sweeping down through Mesopotamia to Egypt in the space of 500 years. Chariots formed the basis of the Hittite empire, inspired the religions of Persia and India, and strangled Egyptian expansionism.

Where things get really interesting is the hypothesised link between the spread of the wheel and the Proto-Indo-European language group. PIE expanded at roughly the same time as wheel technology and it seems like too much of a coincidence for the two to be unrelated... but, of course, coincidences are not how we do history.

The linguists are fairly confident, especially given the number of wheel- and horse-related PIE words that have been reconstructed from studying Eurasian languages, that the invention of the wheel pre-dates the divergence of Proto-Indo-European. What this means is that speakers of PIE had a firm idea of horse domestication and wheels, that the spread of the language not only spread the idea of wheels but likely the physical objects themselves.

Probably. There's a bit of issue here with the Polish and Slovenian archaeological discoveries - the Funnelbeaker culture isn't considered Indo-European until it evolved into (or was supplanted by) the Corded Ware culture. Furthermore, the rapid adoption of the wheel through Mesopotamia is remarkably fast and calls into question the chronology of a single-source expansion model. It would seem the linguistic theory doesn't quite match up with the physical archaeology which is rather frustrating. Some theories have proposed Central European and Steppe cultures simultaneously invented the wheel, others have suggested PIE originated in Anatolia rather than on the Steppe. The debate is on-going.

As previously mentioned, the wheel did spread to China and was certainly known by the first millenium BC. The wheel also spread to Africa through Egypt though it does not seem to have been popular, only being widely adopted for transport during the early days of colonialism. The wheel, despite the popular myth, was certainly known in the pre-Columbian New World but, much like Africa, was relegated to children's toys in Central America (there's no exact date but wheels show up around 1000AD). It's thought the lack of a suitable draft animal (llamas weren't widespread, bison are very grumpy and hard to domesticate) hampered the development of the wheel as a tool for long-distance transport. Much like the aeolipile seems to have hinted at the steam engine and an industrial revolution, the New World civilizations didn't see much need for improving transit when they had plentiful manpower.

Edit: According to this site, apparently the predominant Mesoamerican religious beliefs encouraged people to avoid the use of mechanical tools: "by sacrificing yourself to hard labour you would hope to be rewarded in the next life, and to use a wheel would have been considered worthless." Not sure what the sources are for that but an interesting suggestion nonetheless.