I breed fancy mice and rats and I would be happy to help you out.
1) True hairless mice (aka rhino) are lacking a thymus and thus have a myriad of immuno deficiency concerns that cause them to have (and spread) all sorts of fun diseases. I am working with a lab-strain of hairless mouse now to see what I can do with them, but I am pretty far off from any real results.
2) A "fuzzy" hairless is sometimes confused with a true hairless mouse because it's fur is so short and so fine that it's barely there. The best ones have a sort of peach fuzz about them and are essentially naked, but have no health concerns or issues.
His fuzzy hair is barely there!
Here are some babies!
3) Rex/Double Rex causes curly/wavy fur. A single dose causes a "perm" and a double dose causes a "burnt out" look with bald patches (sometimes). Double rex is more like patterned baldness, and isn't really considered a desirable trait.
Wavy Rex vs Straight Normal furt