After I switched the JS Source on the Compatibility settings submenu to Minified Scoped, the plugin started working again.
Once you resolved the jQuery issues, you should switch back to the “Minified” option, as it's much more compatible than the current one: https://i.imgur.com/Mboo9xg.png
The second issue is most likely related to pages with categories, there must be some kind of mixup. The bug might not be related to the search, but I don't know.
If you can provide FTP access as well, I can try to debug through the queries and check where the problem might be.
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