Rooted (Verizon) Samsung Galaxy S4 -- run FM Radio app?
As far as I knew, the Galaxy S4 doesn't have an FM radio transmitter in the phone, as Samsung opted to leave it out due to "lack of consumer interest".  It is present on the newer models now, yes.

If you root the device, you can flash a custom ROM for you device, yes, and it'll work on Verizon's network (assuming you are flashing a Verizon compatible custom ROM).