Actually is there a way to install the Genesis framework to WP in conjunction with a Brainhost theme and where would I upload it to ?
I guess I’m not clear on what you are actually trying to achieve, so my answer is yes. You can install the Genesis theme and the Twitter Blog 1.0 theme in WordPress at the same time. Can you use them together? No.
You can install as many WordPress themes as you like. (assuming you aren’t being hosted on a WordPress Multi-User site where your host doesn’t allow it ) Just use an ftp client or the file manager in your hosts control panel. Here are the directions: Adding New Themes
Genesis is a theme framework that can be easily changed by the use of many different child themes. That means that any theme designed as a child theme just for Genesis, will not work unless you also have Genesis installed, but it does not mean that any theme will work as a child theme for Genesis.
You can install pretty much any and as many WordPress themes as you like, free or otherwise, but generally speaking you can only use one theme at a time on your blog.