What is the ruling on two people performing the Rami (pelting of the Jamarat) on behalf of two other people (one of whom is elderly and the other remained with him to care for him) during the days of Rami?

The general rule is that the pilgrim should stone the Jamarat himself, and代理 is not permissible except for a valid excuse such as illness or imprisonment.

Therefore, if the elderly Sheikh is unable to stone due to old age or illness, it is permissible for him to appoint someone to do it on his behalf. However, your staying behind to care for the elderly Sheikh is not an excuse that permits appointing someone to stone on your behalf, as you can go stone after one of your companions returns to care for the Sheikh. Therefore, you are in the position of one who omitted the stoning despite being able to perform it, and a dam (sacrificial animal) is incumbent upon you, which is a sheep to be slaughtered in the Haram and distributed among its poor.