The process for the name change is the passport follows the passport renewal application Click here to see the passport renewal application process and we have mentioned the details/documents required for the name change
If this is your first time getting US passport then you need to do a new application with your orginal name proof document (Birth Certificate) and the secondary document to prove your new name (Marriage certificate, divorce decree, or court order). Click here to see the new passport applicaiton process
Yes, you can update your passport with your new married name by following the passport renewal or new passport application process. You'll need to provide the necessary documents to support your name change, including your original marriage certificate.