Different SSL cert suppliers require different things. Each will require, at a minimum, the domain name being secured. Of course they will. A cert is designed to protect a specific domain name, and is issued for that name. So the domain name itself is not private. Why not simply use an e-mail address attached to that domain name?
EV (extended validation) certs give a lot of information about the organization which purchased them. Browsers will highlight sites with these certs with a green address bar or green lock. These are commonly used by financial institutions, as they provide verification of identity, so you know you're communicating with the correct institution. Normal (domain validated) certificates do not. The SSL supplier will probably ask for your name and address for their own files, and for billing purposes, but this data will not be included in the certificate.