You should always install all Windows Updates, this includes IE. However, you are running Windows 8.1, which has IE 11 by default.

Note that just by updating IE you are not going to lose your Firefox.

If you mean Windows 10, yes, you should update when it releases next week. Note that you need to use "Custom" settings in order to keep Firefox as your default browser, if you use Express Settings then you will need to reset Firefox as your default.

Hi, If it helps to get rid of the annoying message from HP, there is no reason why you shouldn't download IE explorer. You probably won't have to actually instal it but - even if you do - as long as you have ticked the Option for Firefox to always be your default browser (see images), there should be no cause for concern over using Firefox. If IE prompts you to make it the default browser, don't do it.

Note that it is recommended to backup your personal data in the Firefox profile folder regularly and especially if you would update your Windows version.

You can install as many different browser as you want, you only need to decide which browser you want to set as the default browser that is used when you click a link in another application.