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I don't think you understand the proper definition of "lean" and what ignition advance affects. Lean refers to your air to fuel mixture ratio. Ignition advance (dumbed down as "timing"; not the best term as there's more than one type of timing) refers to WHEN the fuel mixture is ignited.

If you ignite the mixture too soon the engine soaks up more heat and the cooling system must evacuate it. If you ignite it too late the heat energy exits with the exhaust and the valve train, O2 sensor, turbo, cat must all deal with the extra wear and tear. Bumping the ignition timing beyond 14 isn't going to result in much difference if the O2 sensor checks out I would test the AFM and if that still good you may need to replace the cat. A lower octane fuel may help out as it's easier for the engine to burn... just baby it while it's in the tank.

Make sure the O2 sensor is very well crimped in place and that you clear the codes before getting the vehicle tested again. Do NOT solder the wiring for an O2 sensor. The dissimilar metals may cause a dielectric reaction skewing the O2 readings significantly. The MKIII uses a narrow band sensor which uses very small voltage signals to determine if the mixture is stoichiometric and that random and tiny voltage spike from a dielectric connection could cause the readings to be significantly off.

Do NOT reset the codes before testing.... give the car a day or two of MILD driving to learn the finer fuel adjustments needed for optimal power and emissions compatibility. You are VERY likely to fail on a car that in its learning mode if the sensors are all old and weathered.

The code 51 doesn't mean a whole lot. Code 51 only means that the IDL switch wasn't closed when the diagnostic mode was running, the TPS isn't properly calibrated, you stepped on the throttle or the A/C was on. This code will clear automatically cone the offending trigger is resolved. A worn TPS shouldn't cause you to fail a test completely... it's a minor trim sensor.