According to the mechanic the clutch magnetic plate of our mazda tribute model 2005 burned which caused affected the function of the air conditioning is this possible?

It sounds to me like you need to find a reputable transmission shop.

Question edited for clarity
Question moved to category.

Water leaks don't cause overheating unless there is no, or very little coolant..

Air locks cause overheating.
Failed water pumps cause overheating.
Blocked radiators cause overheating.
Stuck thermostats cause overheating.

https://www.fixya.com/cars/r29872074-vehicle_cooling_system_no_heat

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Mazda Tribute Questions - CarGurus

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Tribute won't start - Mazda Forum - Mazda Enthusiast Forums

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Although I am not a Mazda mechanic, I do have some experience with these problems.
When you get everything fixed, the light will stay on until you have the computer "reset".
This may cost up to several hundred dollars at some dealerships.
If they try to reset it, and something is still wrong, they may still charge you the fee.:-(
God bless your efforts.

Possible causes
- Low transmission fluid level
- Dirty transmission fluid
- Faulty torque converter clutch solenoid valve
- Torque converter clutch solenoid valve harness or connectors
- Torque converter clutch solenoid valve circuit is open or shorted
- Transmission internal mechanical problem

it is a regular fault
generally it is the Heated oxygen sensor (HO2S) in the exhaust manifold before the cat converter
other cause can include engine mechanical fault ( unlikely)---wiring---ignition / fuel system--injectors ---ECT/MAF sensor---ECM
a vacuum problem will affect both banks as will a PCV valve
The HO2S reads the exhaust gas composition and reports to the ECM which then adjusts the air /fuel ration accordingly to keep it within set parameters
The code says bank 2 sensor 1 which is the HO2S sensor so replace it and then have the codes read again

air conditioner has a clutch when the clutch is engaged there will be more of a load put on the motor causing the idle to compensate for the load

It sounds like your air conditioning compressor or air conditioning clutches have gone bad the solution is to have rebuilt (or buy a rebuilt)air conditioning compressor. Do not attempt to work on thr compressor yourself unless you are certified in air conditioning, gas in system can kill you if you breeth it in.

The stop light switch is sticking. If it's a mechanical one that is operated by the movement of the brake pedal the replace the switch. If a hydraulic one is fitted, remove one wire and see if the stop lights go off. If so then replace the hydraulic switch.