I think in the ANKC they are all listed under "Belgian Shepherd" but I don't think people mix types.
Yes, you can mix the varieties. This is called "intervariety breeding." AKC is the only kennel club that won't allow it; It WOULD be allowed if the breed clubs would get off their high horse. The club officials are afraid of losing control and power, and as long as that is the case, AKC won't allow intervariety breeding. People can still do it in the USA if they register with UKC or whelp the litter in Canada and register it with CKC.
However it is very commonly done in Canada and Europe.
Even without intervariety breeding, things happen. Two malinois can produce tervuren or groenendael. Two groenendael can produce tervuren (this is most common). Two tervuren can produce groenendael... Etc.
The varieties are basically the same breed, just split up into different coat types/colours. Malinois are used for working most often so you tend to see more drive and energy in them, but that certainly isn't always the case. The breed, overall, is basically known for similar characteristics between the varieties. The slight differences could also be due to other genes attached to the genes for physical traits; Mal breeders notice that their longhaired puppies tend to be calmer with less drive than the shorthaired pups. And the fact that tervs are known for worse structure and temperaments than groens could be due to some breeders placing colour as a priority (obviously groen breeders don't worry much about colour).
Groens and tervs tend to have a much bigger issue with epilepsy. Mals tend to have higher rates of hip dysplasia (ALOT less testing done in the working lines), and laekens appear to have more elbow dysplasia.
But all in all the same breed, and I hope that it stays that way. I will fight for keeping them as varieties, and not breeds. The gene pool is small as it is, and we can't keep other varieties from popping up in our litters, so it's only fair they remain as one.