Hi!
1. It’s a bit hard to tell, it could be a theme or .htaccess rewrite rule related issue as well. If I simply visit the search results page without triggering the search, with an empty phrase, then I also see the message.
If you want I can investigate this via FTP access, but you can also try increasing the number of characters to 1, so that the empty search is not triggered.
2. You can turn that off here.
3. This documentation section should help you with that.