Hi,
Thank you for the details, it helps us a lot.
1 & 2. Site search terms are triggered whenever the search results page is displayed. Currently you have the ajax search pro configured to trigger specific events. Events are different from that – those trigger on specific search actions.
You can however change the method from Event tracking to pageview: https://i.imgur.com/hfhcFXs.png
In that case whenever the live search is triggered the plugin sends a “pageview” as if it was done via the results page. These should eventually show up as search queries. You can even use custom query variable, explained in this documentation.
3. I am not sure about that, it depends. Is this noindex attribute set somewhere in wordpress or via robots.txt file?
You have multiple options to exclude results. One is to exclude them by categories/taxonomy terms or by tags. There is also an option to exclude them by custom field values. If nothing else is left, it is still possible to exclude them manually.