Only in the US. In the EU, they're registered as a Luxembourg-based bank.

http://www.businesswire.com/news/home/20070515005225/en/PayP...

Yes. They are not a bank, but they are a Money Services Business (in the USA). E.g. see http://paymentsviews.com/2012/08/13/so-you-want-to-start-a-p...