OP, I used to think about that too, but I realized the fact is , if someone wanted to find me they easily or with little effort could. So at the end of day, I could care less that someone can look me up on QRZ or the FCC database.
If you want to be licensed and keep your address private, you can get a PO Box. As far as your name, I highly doubt the FCC will allow you to use John Doe or any other alias. Much like a driver's license, your FCC license has to be your true identity.
Another thing you will have to be prepared to do is show forms of identification to even sit for the exam, that must be verified by the examiners, I had to show my DL, to which three separate examiners had to sign off that they personnaly verified I was who I said I was. I have since found out in other areas this is only done by one examiner, so it depends on the VEs I guess.
My point is, if you want to be a ham, you are going to have put some of your personal information out there for the world to see and folks are going to be able to look that up that you contact on the air. I don't think there is anyway you can get around that and get licensed.