Is it permissible to use air conditioners supplied with electricity by family members through illicit means, and to enter rooms that are still cool because of them, or to operate them at the request of relatives and then donate the equivalent of the consumption cost in charity?

It is not permissible to operate air conditioning units or similar appliances with stolen electricity. However, there is no harm in you entering a room cooled by it, as long as you did not intend to enjoy its coolness and entered it for another need.

As for your obedience to your maternal uncle in operating the air conditioner, it is not permissible, because it is cooperation in sin, and there is no obedience to a creature in disobedience to the Creator. The intention to donate the value of the electricity does not make it permissible to steal it or assist in its theft.

You must pay the value of the consumed electricity to the responsible party, because you directly operated the device, and your maternal uncle has no authority over you in this matter. A rational, mature person who is commanded is responsible for his actions unless the one who commanded him has authority or guardianship.