Hi!
1. In that case, you are looking for the search override feature. It’s worth noting, that some themes might not support this – when there is a custom query executed within the theme files, the plugin won’t be able to replace the results with it’s own.
2. You can use both, if you want to. When Relevanssi is active, the search results on the results page will be coming from Relevanssi, and the live results in the drop-down list will be unaffected.
3. By increasing the minimum character cound to trigger the search to 1 or more: https://i.imgur.com/qXoplTz.png
That should do the trick.
I hope this helps!