When you flash a ROM, you get the ROM you flash, rooted or unrooted. Probably all ROMs, other than stock, are rooted, so if you flash something "close to stock" it'll be rooted. But if you put the Samsung TW firmware in there, it won't be rooted - you'll have to root it. (Root is just one additional file in the system - which you can't save to system storage unless you're rooted, or you use adb to gain root or you use some exploit, which is what all the one-click root programs do.

 

I used Bone Stock for a long time, it worked nicely. It's not longer under development, but since it's a stock ROM, there isn't any reason for it to be...

However, you haven't stated what carrier you're on, and the stock ROM for one carrier is different than the stock ROM for another carrier (that's actually true of all ROMs, not just stock). Make SURE that whatever ROM you choose is for YOUR device/carrier. Just because it says it's for an S4 doesn't mean that it will work on YOUR S4.

This is Bone Stock, for Verizon.
[ROM] BoneStock v1.1.4 (VRUAME7) 8/24/13 - xda-developers