Hi there, @Courtney_Dean 
Unfortunately, it’s impossible to exclude the specific pages from the search result, but we have this idea on the Wishlist. Feel free to upvote it - Exclude specific categories/pages from Search results.
I am not sure if it’s possible to display the out-of-stock listings at the bottom, but I’d be happy to check it with the devs. Could you just send me a link to the page, where your widget is installed?
Regarding this idea, I am afraid I don’t quite get it. Could you please elaborate on what you mean?
The request for the filter option is already on the Wishlist. You can upvote it here 
- Add Filter option