Is what the Shaykh mentioned true, that the angels record the groans of the sick person—"Ah, ah... Allah, Allah"—and that he is rewarded for it? And what is his evidence?

The angels record everything a servant utters, including the groans of a sick person. Some of the Salaf disliked the groaning of a sick person, such as Tawus and Imam Ahmad, who stopped groaning after he was informed that Tawus disliked it and said it was a complaint. Some elaborate on the ruling regarding groaning as follows:

Disliked (Makrooh): If it is due to annoyance or impatience.

Forbidden (Haraam): If it expresses discontent with what is decreed.

Permissible (Mubah): If it is merely for comfort.

Recommended (Mandub): If it expresses humility and neediness before Allah the Almighty.

Everything that afflicts a Muslim of pain and aches expiates their sins and erases their evil deeds.