Q. I would like to know if I can use this plugin for a site wide search, rather than just specific to a post type.
A. Yes, you simply check all post types you want it to include.
Q. How would I get the search term to be posted to the filters on the search page?
A. Whilst with a lot of hacking it might be possible to have a search box on (say) the home page, send results to the search results page, and for the S&F form to them pick that up, it is not easy. Better to make a search form in S&F as just the search box on the home page (with Relevanssi support enabled) and use that to send the search string to the search results page, where another more complex S&F form can pick up the search term.
Q. Also is it possible to build a filter form but move the search box to another area on the page?
A. I am unsure what you mean.