Hi,
Thank you very much for all the details, it helps me a lot.
I’m afraid both issues are related to something else. To test how the plugin affects the results page and errors, I first did a search via the plugin for the phrase “test”, then again after de-activating the plugin. Unfortunately the errors persist even when the plugin is deactivated, as you can see on this screenshot: https://i.imgur.com/p6bToAm.png
A possible cause for issue #1 could be a script related to Google Fast Button implementation, as the error refers to that script source, or something very similar. At least that is what I have found after some googling.
#2 could be another search plugin (maybe?), that is active, or a modification to the core functions.php file, or something within the search.php file in the theme directory.