What I will say needs to be fact check because I never dig much on it since I’m not using w11 so I can be wrong. There is a feature called recall which take regular snapshot on w11.
A quick search about gives different results but it is apparently a opt in feature, so unless you enabled it, it shouldn’t.
If you think it does this from your browser search, try using a second device which isn’t windows based to browse the net and see if its still guesses right
Haven’t read so can’t tell you but you will find info at https://leminal.space/post/11699480
Home assistant has a voice assistant but it isn’t a voice assistant. Also, you are correct it is made to control a smart home (at the moment) that also include music threw external provider (or local files) but you can’t search the internet threw it. You can either install it locally or pay for home assistant subscription which include voice assistant threw cloud.
There is many different instances, maybe try another one