Firefox Browser, Irritating?

That's very strange, IIRC Firefox only asks you after the installation process is over, if you want to make it the default browser, nothing more, and as others have said, the only reason why Firefox would behave like that is because you programmed it to run at startup or maybe some kind of malicious program has programmed to open it at start. You should run an antivirus like MalwareBytes and check to be sure that you don't have some malware installed.