B737-800 Captain reported engine failure along with FADEC and reverser lockout malfunctions during climb that resulted in a return to the departure airport.

Narrative

I was the Captain and Pilot Flying (PF) of Aircraft X on Day 0 along with FO from ZZZ to ZZZ1. Pushback; taxi out to Runway XXL at ZZZ was uneventful and normal. On climb out at approximately 9000 feet with the autopilots engaged; I noticed and felt the #2 engine rolling back and the REV amber light came on. I do not recall even getting a Master Caution though. I brought it to my FO's attention and announced My aircraft". I turned off the autopilot and autothrottles so that I could have a better feel as to what was going on. The FO [requested priority handling] and he advised ATC we had something going on with our #2 engine and that we would be leveling at 10;000 feet to run some checklists. The engine was somewhat cycling or surging if the throttle was not at idle as the FADEC (Full Authority Digital Electronic Control) took control. I remained Pilot Flying while my FO went to the QRH checklist for REV (Reverser unlocked in flight) checklist. We both agreed that was the appropriate checklist and we accomplished that first. With the current indications and responses from the engine; we were directed to the "Engine failure or Shutdown checklist".That checklist was accomplished and we shut down the #2 engine and followed the remaining procedures. We still had some time before getting a returning vector back to ZZZ so I transferred control to the FO and accomplished the "Non routine landing considerations" items Telling the FAs (Flight Attendants) our plan; telling the people a short summary; and finally sending the ACARS message to Dispatch. ATC had already been told we wanted Runway XXC in ZZZ and we began our vectoring back to an uneventful overweight single engine landing in ZZZ. As advertised; safety equipment was waiting for us as we turned on Taxiway 1 and stopped facing south on Taxiway 2. I made a PA to the passengers that we would be stopping for the safety equipment to give us a quick look over and to remain seated until we were back at the gate. We asked the Fire detail to look at our #2 engine for any visible leaks or damage which would prohibit us from taxiing to the gate. They indicated that everything looked ok and at that point we taxied to Gate XX. Parking and shutdown was uneventful and normal. I made an entry in the logbook. Maintenance met us shortly after and queried what we experienced and what we saw with the engine and told us they'd be taking it to the hanger."