<blockquote>I guess that host needs to install Joomla in order for it to work?</blockquote>
this statment shows will be in for a tough time and quite a learning curve, Setting up an ecommerce site is a bit more complicated than the standard information type of website.
You should start with the quickstart package and build on that.
I also recommend that you build it in a sub folder of your domain and then you can just move it up a level.
if you do build on a different domain and then want to move it, its easy, you can just use joomlapack, i;ve got it down to a point that it just takes minutes.