Welcome to the 4x4wire forums... I'm sure you'll find lots of info and folks here that share your interest in Toyotas...

Yes the 1135 and 0125 codes are related...

The 1135 code indicates the heater circuit in your front O2 sensor is no longer working and therefore the O2 sensor has failed.

The wording of the 0125 DTC error description is somewhat of a red herring, but correct... What it means in your case, is that after the engine has warmed up and should be in closed loop mode, the computer is not seeing a valid the O2 sensor signal..

Other than checking to make sure the wiring is intact to the front O2 sensor, you need to replace the front O2 sensor... Make sure the dealer confirms your vehicle is California spec and gives you the proper O2 sensor.. The 1135 (vs the 0135 error discussed in the article below) code indicates your vehicle is a California spec vehicle.

Replace the O2 sensor and make sure it's no longer setting any codes.. then I'd recommend you clean the Mass Air Flow sensor as mentioned in the article below.. There's a link showing you the procedure...

Read this article first, and then this article on cleaning the MAF and other items you should check.