A few days after entry to the US as a K-1, you can apply for a SS card (up the 75th day after arrival). Immigration will have you listed by your maiden name, so that is what you must use for your initial SS card. Some K-1's have had success getting the initial SS card name changed to married name by showing your marriage certificate. Most have not had luck doing that, so it just depends on the local office.
As for EAD and green card, whatever name you use on the applications will be the name that is on your cards. If you use married name when applying for EAD and green card, they will be issued with your married name. Also, you can just check a box on the I-485 and I-765 petitions that will allow an updated (married name) SS card to be mailed to you. That is, by far, the easiest route to take. For me personally, there was no rush to get a SS card, so we didn't bother getting one until we filed I-485/I-765. I avoid government offices whenever possible.