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Well the notoriety of buying it didn't exactly help Shkreli's situation much, did it? If you want wealthy people to take an interest in buying it, it seems sensible.

To prevent the buyer from becoming a target for crime.

If people know who has it, they might try to extort him or worse.

I don’t see any reason the information needs to be public.

That would turn off a lot of buyers because it would bring undue attention. Buyers may not know the exact origin of stuff they're buying. A car for instance could formerly have belonged to a person you might not want to run into. A public record of that purchase however would make it a possible target not just for the original owner, but also any rivals he/she may have.

Presumably because shrekli purchased it under those conditions. He can't sell you the right to state the price because he doesn't own it.