Theres many pre-packaged shopping cart solutions which will meet your requirements.
The one we use and support most is osCommerce (free). There are many others that are used by other hosting companies (ie Miva and Angora). All are good solutions and each have their advantages.
I cant speak for the others (as Im not that familiar with them), but osCommerce has all the capabilities you mentioned above (as well as many more). You may not need to buy a template - most hosts offer a free site builder with their hosting packages (though, Im not certain if 1&1 has this offering)
I am sure (well almost sure) you can run osCommerce on 1&1's free offering. I hear their support isnt great (but dont have any first hand experience of this), so you may need to install this yourself - though I would recommend contacting them to find out if they will help you install.
You can download the latest version of osCommerce from www.oscommerce.com
Hope this helps, and best of luck with your venture.