I started a new ticket to send you FTP details.
I found the reason of the jqery errors. It was the plugin called exit intent. After deactivating the errors are gone.
But the problem with the wrong categories still exist.
Many thanks for your offer of having a look at the queries.
Thank you for the correct information. I have found the bug in the search code. In some cases incorrect category IDs were returned, it caused the confusion.
I have corrected the code and adding this fix to the upcoming version. It should be working correctly now.