Is it permissible for someone to stone the Jamarat on my behalf and on behalf of my wife due to her illness, and is it permissible for a two-year-old child to enter into a state of ihram in his regular clothes and for someone to stone on his behalf while he remains with me throughout the Hajj?

It is permissible for a wife to appoint an agent to stone on her behalf if she is ill and unable to stone herself. As for the husband, it is not permissible for him to appoint an agent as long as he is able to stone. If he appoints an agent while being capable, he must offer a sacrifice (dam) to be slaughtered in Mecca for the poor of the Haram, based on the Almighty's saying: "And complete the Hajj and 'Umrah for Allah." This is unless he fears harm to his wife by leaving her, in which case he is excused for appointing an agent and nothing is required of him.