LiamGo

If you get your vaccinations before your Knightsbridge appointment, then they are all signed off on an official immigration form by an approved Panel Physician. Any shots received later have to be signed off by a Civil Surgeon on Form I-693. USCIS does not take documents signed by your GP as proof of immunization completion.
https://my.uscis.gov/findadoctor Find a civil surgeon
Knightsbridge is notorious for giving incorrect info to K1s about the AOS requirements.

Once you have a Social Security number (doesn't matter through what), it stays with you for life and doesn't change. You don't get a "new" one. You can update your immigration status with SSA whenever your status changes. Since you are now a permanent resident, if your card has the "valid only with DHS authorization" text on it, you can get a new card without that text.