No, that isn’t how WordPress works. WordPress uses the theme to combine data from posts and HTML from the theme to create new HTML pages on-the-fly.
The theme is more than just visual styling of content – it also manages the “output” and structure of your content (i.e. your typical HTML markup). So yes, you do need a template, but it doesn’t need to be visually styled. Check out the _s theme.
Now as to pulling in content from another site, consider using an iframe to do this. But read up on iframes and the issues involved with them. It’s always “dangerous” to rely on third-party content which you have no control over – if they get hacked, that iframe could potentially become a security vulnerability on your site.
Ok, so how would I mimic this site? http://www.floridascreen.com/. I want to create something very similar to it using wordpress.