Is it permissible to expose the 'awrah (private parts) of the husband's grandfather to care for him and change his diapers when necessary, and when there is no other caregiver for him? And is the daughter-in-law obligated to do that?

Serving the husband's grandfather is not obligatory for his son's wife; rather, it is obligatory for his sons, and they should marry him to someone who will serve him. Cleaning him in a manner that exposes his nakedness is not permissible except for his wife. If she is not present, one of his male sons should clean him while lowering their gaze. If that is difficult, then it is permissible to look at and touch the 'awrah to the extent of need due to necessity. It is also permissible for his mahārim—including his sons' wives—if there are no men available to serve him. As for a woman serving him in ways that do not expose his 'awrah, such as washing his clothes and preparing food, it is not obligatory in principle, but it is a kindness that is good to perform, especially if requested by the husband.