USCIS replied on 13 November stating:
"The status of this service request is:
You contacted USCIS to provide your updated address. We updated your address to:
XXX X XXXX, USA"
This response seems like a non-sequitur since the purpose of the service request was for "Non-Delivery of Permanent Resident Card". Hoping to see an update there soon.
As for the Social Security issue, I called again today and the rep said that we should just re-apply for a social security card because it's been too long and they can't find my wife's name in the database. I have no idea what could have gone wrong here, but it is what it is.