B737-700 Captain reported experiencing autoflight anomalies related to failure of the TAT probe climbing out of FL200.

Narrative

Climbing out of FL 200 with the Captain as Pilot Flying; the aircraft unexpectedly went in Control Wheel Steering (CWS). We were assigned a level off at FL 230. Attempts were made to re-engage the autopilot to no avail; so a manual level of at FL 230 was made. Since initially the autopilot would not engage in either A or B; we informed ATC we could not continue into RVSM airspace. Multiple information was missing from the VNAV page in the FMS. The TAT was giving erroneous readings. There was no SAT. There was no TAS on the Captain side. There was no pitch bar on the Captain side. No N1 limits were displayed only green dashes. Using ACARS we informed Dispatch of our issues. They responded saying we should run QRH page XX.X FMS FAIL. We didn't have that indication. We only had a VNAV disconnect. We also informed Maintenance to meet the jet in ZZZ. We had sufficient fuel and weather was not an issue. Since the First Officer had better information on his displays; I transferred controls to him and he landed without further incident. ZZZ Maintenance found a fault code on TAT probe. After clearing the fault; they said we were good to go to ZZZ1. Immediately after take off the same problems happened again. The Company was quickly informed since we were so close to ZZZ for a possible return. The decision was made to continue since we had plenty of fuel for the short flight and weather was VFR. We again elected to not go into RVSM airspace even though this time the autopilot seemed to work on the First Officer's side. Dispatch was notified through ACARS again. They asked why were were not using page XX.X in the QRH. We didn't use that checklist because we didn't have that indication; so it didn't apply. The issue was written up again after landing in ZZZ1.Our main concern was who in Dispatch was telling us to run a QRH Checklist that didn't apply to the situation.