Browser always need a internet connection to properly work. So unless you choose the browser option "Offline" that's how you use a browser with internet connection. This is not a official support request forum just a user help forum. And Firefox update can't be turned off or disable unless you use 3rd party software but that's at your own risk to do that.

Hello

About fake notification "Update Available - download now"? what do you mean precisely

About DisableAppUpdate https://connect.mozilla.org/t5/discussions/upgrade-notification-spam/m-p/64080/highlight/true#M22529 for information purposes