Does the prayer of one who slept through the Fajr prayer due to the coldness of the water, and prayed it after sunrise, become void? And is it permissible for him to perform tayammum when he is able to heat the water?

This action is a great sin and error that necessitates repentance, because intentionally sleeping through prayer after its time has begun, with no hope of waking up before its time expires, is considered a major sin. The duty was to heat the water for ablution, and it is not permissible to perform tayammum in this situation, nor to delay the prayer until sunrise. The prayer after that is valid, but it is a make-up prayer (qada') and not performed on time (ada').