Well, they should and they do. But it's like sex - both involved parties should take care of safety. :D So if you care about your site - take care of backups yourself.
ID card or passport copies and if the client chooses to pay for his services by credit card, a credit card scan may be requested, too. And phone verification (always).
I don't think ditching PayPal is a good choice. However, it mostly depends on your customers. Personally, I'd never drop PayPal. As for myself, I also prefer to pay via PayPal than use my credit card directly.