I am self hosting my version so that is where my screenshot came from.
If I were looking for a public instance to use that has this functionality, I'd look to some of the European hosts as they are less likely to use Reddit as their de facto platform relative to Americans
Not in my experience. I am running a docker container and it is one of +10 containers running on my server which is basically a laptop from a few years ago. So far no issues.
They don't keep a search history. Your searches aren't tied to anything, because they aren't even saved. They don't have any reason to save them since they aren't selling any targeted ads.
This was my understanding as well. It's the point of paying. However , it is just "trust me bro" right? It's not like they've been audited for not storing user data? Even just basic logging too could hold user info. Honestly no idea with paid search engines, I know so little.