Hi,
Thank you for all the details, it helps a lot!
1. Depending on the number of items to search from. Generally, a few hundred posts should be okay. But if you have a lot of tags, lots of posts, then it could give more relevant results. I suggest trying it, and choosing based on your preference.
2. Unfortunately no. It is a free API by google, it accepts a very few arguments, and it is very generic. In exchange it is blazing fast, as it uses the google servers, but unfortunately it cannot be customized.
3. Sure, you can turn off all the triggers to display the live search results here: https://i.imgur.com/YJBjM8X.png
I hope this helps!