Marie, I run several WordPress sites, all hosted on DreamHost. I use the VPS service, but started out with Shared Hosting.
You can pretty much do whatever you want with your WordPress install. Some of the sites I manage have been heavily customized to make WordPress do all sorts of things.
Yes, you can use all of the free WordPress themes, you can buy your own and upload them, or you can even start with a Bare-bones theme and build your own (my personal preference).
It will be helpful to have some FTP/SFTP software (FireFTP is a great free plugin for Firefox)