I wouldn’t count on the logo changing. Maybe on newer Android devices, but definitely not in IOS.
Facebook has always been weird for me with links. Sometime they open in the app and sometimes they open in the browser. I am personally logged into facebook in both the app and the browser, so I don’t get the login page, but it is annoying when the browser opens instead of the app. I really don’t have good answer around this. You can build links that will deep link into facebook, but they are different for IOS vs Android. The more I googled for an answer, the more confusing and complicated it all seemed. If you really need something, I would maybe explore a url shortener, like Branch.io, which appears to possible handle deep linking. I’m not sure though. Sorry I dont’ have a better answer.