I can see this breaching their ToS and would probably be shut down by them if brought to their attention. However, what law would you be breaking when you give a private company a fake name.

Regretably, in the US the "Computer Fraud and Abuse Act" (CFAA) makes it illegal to access a computer without "authorization". Courts have interpreted that to mean that accessing a site when you are in violation of their terms and conditions (for instance, with a fake name) is a violation of this law and thus a felony. Learn more: https://www.eff.org/ja/issues/cfaa

New question: does it matter if it's illegal if it's 1) moral and 2) you're anonymous?

If its illegal, then it's much more likely to be shut down. Do you want to release some information once off, or do you want a reliable Web host.