The UKVI will have your passport while your application is processing, so you won't be able to leave the US. You can't apply until after you have your marriage certificate in hand. After you receive your visa, you'll have 30 days to enter the UK.
Yes, you need to pay the NHS surcharge. This needs to be paid prior to applying for your visa (just before) as you have to include you payment confirmation on the application.
Does your fiancé meet all the requirements to sponsor you?