Evening folks, just a lil update. Check was cashed on 7/29, received paper notifications on 8/1, and access code arrived 8/4.
Just today, we went to SSA and was able to successfully apply for spouse's SSN without any issues. Documents we took were passport, recent I-94 printed from CBP website, birth certificate, and marriage certificate. The employee who helped us said the number should be generated within 2 days and card received in 2 weeks!
Still waiting on biometrics appointment.