Do the limits on interaction between a man and a woman fall away if the man is elderly and the woman is young? And is it permissible for her to speak to him respectfully or hold his hand?

The mere old age of a man does not change the boundaries of lawful interaction between the sexes. Rather, it is essential to be safe from temptation (fitnah) and for neither party to harbor desire for the other. If these conditions are met, then there is no harm in speaking to him politely about permissible matters, and for each to look at the other—like non-marriageable kin (mahārim)—without shaking hands or touching. As for looking, whoever has lost their desire due to old age, impotence, illness, castration, old age, or a eunuch who has no desire, their ruling concerning looking is the same as that for mahārim, based on the verse: {or those male attendants having no sexual desire}. As for speaking politely about permissible matters, if looking is permissible, then speaking is permissible a fortiori. Most jurists have permitted hearing a woman's voice if there is no fear of temptation (fitnah) and she does not speak in a seductive manner. Al-Ghazali stated in Al-Ihya': "A woman's voice, other than in singing, is not considered 'awrah (something that must be concealed), for women in the time of the Companions, may Allah be pleased with them, used to speak to men in greetings, seeking fatwas, asking questions, consulting, and other matters." As for touching, the temptation (fitnah) associated with it is more severe than with looking, so it is not permissible except out of necessity and with safety from temptation. However, if there is no safety from temptation and there is a possibility of desire, then the boundaries remain as they were.