Does it necessitate an expiation for appointing my brother to perform the stoning on my behalf after I had already stoned the Jamrat Al-Aqabah (the large pillar) on the Day of Sacrifice, due to overcrowding and fear of missing the time, knowing that this was before the issuance of the fatwa permitting stoning after sunset?

Throwing the pebbles is an obligation in which substitution is not permissible except for an excuse. The excuse mentioned in the question for delegating another to throw on behalf of the questioner is not justified. Therefore, a damm (sacrifice) is obligatory upon him, to be slaughtered in the Haram and distributed to its poor. As for throwing on behalf of women, it is valid due to the excuse of overcrowding.