Center Controller with ADS-B question
What equipment do you have to have for ADS-B? Mode A/C transponder
  only or do you also need Mode S too? Is ADS-B tied into Mode S or are
  they completely 2 different components?


Either a mode S transponder with Extended Squitter (the ES broadcasts the ADS-s data) which broadcasts on 1090 mhz or a UAT (universal access transceiver) broadcasting only ADS-B data on 978 mhz. Both transmissions are received by FAA ground stations and the aircraft is placed in the ATC computer system where you see the ADS-B info.


  Do you need a Mode S transponder to fly? (I think from what I read
  Part 121 ..commercial you do, but not required for GA if inside the
  US)


yes, the advent of ADS-B does not change any requirements for transponders. Someday maybe but not now.


  What is an Example of an ADS-B "code" ? ....to clarify...If it is
  registered to the tail number, I would assume you have to register a
  specific "code" to tie the 2 together.


All ADS-B transmissions must output the same transponder code value that the pilot has dialed into the transponder. All ADS-B equipment has some sort of interface the determine Transponder code and that code goes along with the ADS-B transmission.


  Is the ICAO hex number the same as the ADS-B number/code?


ADS-B equipment broadcasts transponder code, ICAO hex, and the pilot's choice of flight ID or N number. There is something called anonymous mode and I'm not sure what's sent while in that mode.


  I ask about ICAO hex because I see some ADS-B equipped GA with hex
  numbers in the system and some do not have hex numbers but still
  verified ADS-B equipped. I want to know if they are both needed!!!


I've installed approx. 25 ADS-B units in small GA aircraft and every configuration I've seen has a entry for ICAO address. It's possible that you've seen improperly configured units or maybe there's something about it I don't know.


  @jScarry> I’m confused about your premise. Why would a pilot call ATC
  to verify
  
  
    that their ADSB is working?
  


The A in ADS-B stands for automatic so it works without any control input from the pilot whatsoever. Its just always on without having to actually turn it on. Consequently, he has no real idea whether or not it is working other than the possible absence of a fail message... so he calls ATC to ask him if his ADS-B is showing up or not. Most likely this is his first flight after the install and he just wants to know that he's showing up and that his $7k is actually doing what it's supposed to.