A300 pilot reported the aircraft had unreliable airspeed indications during cruise. Flight crew returned to departure airport and during taxi the tiller wheel jammed requiring a tow to parking.

Narrative

Cause of the [priority handling request] was Unreliable Airspeed on the captains side. Before we left; the following deferrals were listed in the flight release:30-XX-XX-X NO. 1--TAT PROBE HEATERS: (NO. 1 PROBEINOPERATIVE)33-XX-XX-X FLAP PEDESTAL BACKGROUNDLIGHTING--COCKPIT/FLIGHT DECK/FLIGHT COMPARTMENT AND INSTRUMENT LIGHTING SYSTEM ? INDIVIDUAL COMPARTMENT AND INSTRUMENT LIGHTS (EXCEPT F/O DOME LIGHT; GLARESHIELD LIGHTS; STANDBY ATTITU(AFDS MODE CONTROL PANEL INTEGRAL LIGHTING MUST BE OPERATIONAL FOR NIGHT OPERATIONS.)32-XX-XX #1 - POSITION DET SYSTEMS21-XX-XX-X #1 - CABIN PRESSURE AUTOMATIC CONTROL SYSTEMS: (ONE INOPERATIVE) ? ASSOCIATED SYSTEM DEACTIVATEDWe stepped to the jet only to find Maintenance working on a panel that was missing from a previous flight. They had a scissor lift up and were drilling holes for the brackets that were missing to reattach the panel. There was no log book so we requested and received clearance to start the preflight.Just prior to level-off at FL340 we received multiple issues all at the same time. The F/O Pilot Heat light was illuminated with no associated Master Caution. The Pitch Trim 1 released to OFF. The Captains Magenta speed bug disappeared and the magenta commanded airspeed went to dashes. A few minutes later; we received a Master Caution for ANTI ICE - CAPT TAT HEAT FAULT -ADC 1 IND - Monitor. We ran the ECAM and restored Pitch Trim 1. The F/O Pitot light extinguished on its own and the Captains Airspeed Bug and Commanded Airspeed came back on their own. At this time I noticed that the Captains #1 PDF was flickering randomly to a completely black screen and back on again immediately. These were random and were not associated with the other issues we received at the same time. This continued through the duration of the flight. We then had all the previous issues happen again at the same time. F/O Pitot Heat light - Pitch Trim 1 - Capt Airspeed bug and Commanded airspeed. I monitored the airspeed and noticed when the Captains Airspeed Bug and Commanded airspeed restored on its own there was a +/- 5 to 10 fluctuation in the indicated airspeed. This ultimately drove us to the UNRELIABLE AIRSPEED checklist and to [request priority handling]. The F/O did not have these fluctuations in airspeed and remained steady the whole time. After the Captains airspeed fluctuated it would settle to the same airspeed as the F/O.I transferred aircraft control to the F/O and he flew the remainder of the flight. He maintained a safe; wings level airspeed and altitude. We had these same 4 issues happen a total of 5 times. Each time the Captains airspeed would fluctuate momentarily and then settle down the match the F/Os. When we initially had these issues we were able to contact dispatch on the radio. Soon after we lost communication with them. This proved to be one of the most frustrating issues - trying to re-establish comm with the dispatcher and maintenance. Communications provider is a disaster! We were able to call via ZZZ and establish the link; but as soon as the dispatcher would talk it would go to complete squelching. We tried multiple times with no success. We broke the connection and then re-established with no luck. We received an ACARS to try ZZZ1. We tried - with absolutely no luck - the exact same issue. We then were able to reach another communications provider who was able to establish communication with the dispatcher. This process of trying to contact the dispatcher via the communications provider took over 30 minutes. ACARS messaging is simply to cumbersome and distracting when you are trying to monitor aircraft systems especially with an airspeed issue.We were able to coordinate an Air Turn Back with dispatch for ZZZ and coordinated with ATC to return. Arrival and approach into ZZZ was uneventful. Upon landing the Amber REV UNLOC did not happen immediately when the F/O cracked the Reverse Thrust. The #2 then came on and slowly went to Green REV. I stated no reverse left engine and the F/O stowed the reversers. At approximately 50-60 knots I took control of the aircraft and continued to slow. At approximately 15-20 knots I imitated the turn onto the Highspeed taxiway 1 off of runway XX only to find the tiller wheel completely jammed and the aircraft not responding to any movement left or right. I brought the aircraft to a complete stop about 3 feet right of centerline. We relayed to tower that we would not be able to taxi clear. We started the APU and shutdown both engines. We then coordinated with tower; crash fire rescue and ramp to be towed to parking. After the tow bar was hooked up and when maintenance selected the steering bypass switch the rudder pedals fluctuated left and right like there was a huge release of pressure. The tow to parking was uneventful. Maintenance has reached out and they have found that the Landing Gear Proximity sensor was bad. Maintenance informed me that it was sending mixed singles to different equipment telling some that we were airborne and others we were on the ground. They have since replace the Air ground sensor; Flight control computer; MCP; Pitot Heat Relay and TAT Relay.