Rooted (Verizon) Samsung Galaxy S4 -- run FM Radio app? I'm considering rooting my (Verizon) Galaxy S4, currently running Lollipop as pushed by Verizon in August 2015 (plus any updates that have pushed out since then).  There appears to be no likelihood that Verizon will release an update that lets me use an FM radio app such as NextRadio on this (three year old model) phone, despite the needed hardware being present (and Verizon now allowing FM radio on newer Samsung models, S6 and S7).

If I were to root this phone, either with Lollipop or Marshmallow (providing I can find a ROM for the newer Android version), would I be able to continue using it on Verizon's network, as well as use FM radio?  If so, what do I need to watch out for?  I can't (at this time) afford to replace the phone with a newer one, so it's critical that it continue to work as a phone with the existing SIM card, regardless what I do to it.
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).