Wow, that does sound really supicious. When I signed up with dreamhost it was nothing like that, there should be secure server pages (allways verifable by the lock symbol at the bottom of your browser page). And I would never make an imprint of my credit card, that's just rediclous! there is no need for that what so ever. The latest systems use the three digit security identification number on the back of you credit card as an extra step, but no one needs to see an imprtint of your card or any other kind of reproduction.
I just went to dreamhost.com to pretend to sign up, and I got a secure page at panel.dreamhost.com. Is that what your browser is displaying?
You might go to this page and let some one who works at dreamhost know abotu this.... http://www.dreamhost.com/contact.cgi
I would recommend checking your computer out throughly with an Anti-Virus Program, Spy-Bot S&D, and Add Aware. Let thoes program throughly scan your system, and then try again.
I'd be curious to know how this comes out...