If you want to work, you need SS number. No sure about your timelines, but if there is less than 15 days on the I94 than you need work permit before SS card is issued.
The longer you wait the longer it takes. Hopefully you get work permit soon after filing. My wife still waiting for green card but at least has work permit and state ID
Please complete your timelines
Still never heard of exemplified certificate. When my now wife and I married we went directly to court house to get certificate that was signed by officiant and we've not had any issues.