if you order something online via PayPal, can the person see the name of the person who purchased it on their credit card?
On the systems I've seen, yes, the customer name (and sometimes billing address) is visible.
The paranoid's solution would be to purchase prepaid cards, which do not have a cardholder identity associated with them. I don't know what info they do provide.
Usually yes, as that is the information given to them, technically they aren't legally able to use that information in any way, or sell it/give it to anybody else.
It's just a verification measure. If someone were to steal your credit card and use it, with their name rather than yours. When you go and make a claim against a fraudulent transaction, that information would show up and then make claiming your money back much easier.