Hi stuart ,
Regarding the chat opening automatically,
This could be caused by Facebook Messenger remembering the previous opened state.
The chat widget remembers if the user has opened the chat window and it will automatically open until it is closed again.
Regarding private browsing mode,
Facebook Messenger doesn't work in private browsing mode in Edge since Edge has a built-in Tracking prevention that blocks Facebook's scripts when set to Strict.
Private browsing mode is automatically set to Strict, that is why it's not appearing.