Pilot reported an FMS failure in flight and that the FMS was responsible for multiple systems failures and disregarded it's information; relying on the other; operational FMS. The malfunctioning FMS was placed on MEL after reaching the destination airport.

Narrative

We were scheduled to fly Aircraft X. When we got our paperwork we saw the CAT1 downgrade due to a former write up of LVL 2 STAB OUT OF TRIM WITH AP2 but we were told that issue had been fixed. It was my leg that morning and we were to climb to FL 380. With the cost index given; we were climbing at 340 kts. At around FL 270 we encountered some moderate turbulence and I lowered the cost index to give us a slower speed for a good turbulence penetration speed. The aircraft now showed a .82 climb mach speed. Climbing through FL 340 we received a LVL 1 CABIN PRES RELIEF message and I called for the QRH. At this point we noticed we were climbing at around 3300 fpm. We then received a LVL 1 AVNCS FAN OVRD. Now passing through FL 360 I noticed were climbing at 4000 fpm and our speed was rapidly decreasing to about .77 mach which was right around the foot on the speed tape. The aircraft began a level off and the power started to reduce. I quickly moved the thrust levers forward to keep the thrust at max and I put in a lower VS. The tape stayed blue but was almost touching the flight director. The aircraft leveled off at FL 380. I was prepared; with my hands on the controls; to do a stall recovery maneuver; however the aircraft was just able to power out of the slow; behind the power curve; airspeed the autopilot had put us in. Once we felt we were safe; I switched to AP 1 and the rest of the cruise portion went uneventfully. Then on descent in to ZZZ1 we noticed that my FMS showed that I was low on the vertical path while the CA (Captain) FMS showed that we were high. There was a 2000 ft. difference between our FMSs. We both started to do the mental math to make sure that we crossed each fix at the required altitudes while we tried to figure out whose FMS was correct. We determined that the left FMC was more accurate. We discussed the issue with maintenance on the ground and we said we believed that this was all related. The AP2 seemed to not be getting good information from the FMC and therefore could have been the reason for the previous trim issue. This could have caused the level off issue; and then our descent issue later on. ZZZ1 maintenance deferred FMS 2 and we flew the aircraft to ZZZ3.