We shouldn't have to convince you. It is a personal choice you have to make, much like jailbreaking. http://www.androidcentral.com/rooting-it-me-some-qa
^that has a bit of background as well as some info on how to root and un-root.
This is something in which you have to do the work to find out whether or not this is the right thing for you.
i use synergy which is basically stock mixed with note features like multiwindow. Not really sure how it is on sprint but I posted a link to XDA where you can find everything you need to know and lots of people who could point you in the right direction if you can't figure it out yourself. look around and make sure you have an understanding of what you're doing and if you need it first. lot of cool stuff you can do and everyones opinion on what is the best is different
I use Jelly 'Beans' 15 Custom ROM. But it might only be for Verizon. Do some Google'n and you can find a ROM that fits you best. Most of them have pictures and videos you can look at. Just make sure it works for Sprint.
Yes you should root. It'll give you many more things to do with your phone including but not limited to: custom ROM's, theme's, root only apps, etc. Even if you just stay on a rooted stock rom it'll still be better. Not to mention if you run a custom rom it'll be faster since developers tweak the hell out of it for speed