Hi Colin,
1. I believe there is this option in Litespeed cache. That will recognize user agents and make separate caches for different browser types. That should resolve the issue.
2. By default the plugin redirects to the results page, that redirections seems to be all right. I see the same exact results for an empty query (by entering a space character) to the search bar.
These definitely exist as the plugin requests them via a WP_Query, but I suspect those might be draft or trashed or private items maybe? Check if the post statuses are correctly set on the search settings or on the index table panel if you are using that.
If you are using the index table and these items were trashed programmatically and the save_post or wp_insert_post hooks were not triggered, then the index table can’t update/remove the items. In that case try re-creating the index, that will solve the issue.