Is it permissible to spend the night outside Muzdalifah by 200 meters and return home on the Day of Sacrifice, and to appoint someone to stone Jamrat Al-Aqabah due to the illness of a child and the exhaustion of the wife, and to spend the night in Al-Jamarat and stone the pillars after Fajr instead of after Dhuhr?

Your staying outside Muzdalifah due to overcrowding necessitates a fidyah (expiation): a sheep to be slaughtered in Mecca and distributed among its poor. This is the more cautious approach, though some scholars are of the opinion that there is no penalty upon you as there was no negligence on your part. As for the ramy (stoning of the Jamarat), you erred by appointing someone to stone on your behalf while you were capable of doing it yourselves. Your stoning of Jamarat Al-Aqabah before dawn on the eleventh night is permissible. However, stoning the Jamarat before Zawwal (midday) on the first of the Tashriq days is impermissible according to the majority of scholars, and therefore, you are required to offer a hady (sacrificial animal) for abandoning the stoning. Regarding staying overnight in Mina on the eleventh night, if you did not spend most of the night there, and if you had a valid excuse, then there is no penalty upon you; otherwise, you are required to give sadaqah (charity).