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The check engine light codes are more than likely completely unrelated to the noise. The noise could either be internal transmission or even possibly wheel bearing related because it increases with speed.
The failure codes are definitely different.
P0171 is a system too lean code meaning that the air fuel mixture in the engine has excessive air, therefore running lean. There are many things that can cause a lean code such as intake leaks, bad mass air flow sensor, defective oxygen sensors, ect.
P0125 failure in bank 1 sensor 1 oxygen sensor.
P0135 failure in bank 1 sensor 1 oxygen sensor heater circuit
P0141 failure in oxygen sensor heater circuit
I would begin by replacing the upstream oxygen sensor and then inspecting the intake manifold and throttle body for air leaks by spraying throttle body cleaner around it and see if the idle changes.