Can I make Firefox the default browser in iOS
You cannot change the default browser (Safari) unless you jailbreak, which you do not want to do. You can however use supported apps. For example, just like how ads in browsers may open app store "sponsored" apps, YouTube will open Chrome if enabled to. If the app does not specify which browser to launch in when a browser-required link is clicked, Safari will be launched.

If programmed to do so, apps are well able to launch (and transmit data) to other apps. You cannot change Apple's iOS default settings but you can decide which apps you use such that the apps you like will be supported.

Update (2019) : The "Share" option in iOS may be able to launch specific browsers, eg: Firefox has this option.