Is it possible to set Firefox as the default browser in iOS? Is it possible to search something using Firefox in Spotlight search? When I choose the option "search in the internet" it always open Safari. My question is then, is it possible somehow to set Firefox as the "default" browser in iOS?

I am using iOS 9.1 in an iPad mini.
No, setting the default browser is not supported in iOS. You have an option that I do not recommend, which is jailbreaking: http://www.idownloadblog.com/2014/02/02/make-chrome-default-browser-ios/