Diamond grit for polishing and grinding is now a cheap commodity. In 10,000 ct. lots, I pay from 5¢/ct. to 30¢/ct. depending on specific grit properties. I haven't searched for it, but diamond sandpaper should be a thing at these prices.
Unknown at present (nobody sells diamond sandpaper yet that I know of), but not likely. Diamond is as bioinert as gold, fluorocarbons, and similar materials in not provoking inflammatory responses when implanted in tissue.
Your point about unexpected behavior of dust is a good one. AFAIK, the worst dust hazards (aside from outright chemical or radiological toxicity) are from spiky, acicular particles like asbestos. Diamond dust is either well-formed small crystals or blocky fragments, depending on how it's made. It doesn't exist in needle-like forms that trigger pathological responses from otherwise nontoxic materials.
Diamond abrasives are definitely a thing. But for sandpaper, you rarely need anything more than the typical ceramic or silicon carbide abrasives. They pretty much are relegated specialized tooling for polishing, cutting, or abrading ceramics, concrete, tile, etc.
