Is it permissible for a patient with a terminal illness to pray without performing ablution (wudu) or tayammum, by simply patting his hands and wiping them while in bed due to an inability to perform wudu, and can he combine the Dhuhr and Asr prayers by delaying the former?

It is not permissible to pray with tayammum (dry ablution) except when water cannot be used for fear of illness, or its aggravation, or delayed recovery. If tayammum is permissible, merely shaking the hands and wiping is not sufficient; rather, it has a known procedure. Combining Dhuhr with Asr prayers during the time of Asr is permissible due to hardship in such a case.