First off, congrats on your upcoming nuptials!!
My husband and I just went through the same thing. After reading all the advice on this forum, we got married in November at the courthouse in Ohio and applied for the spouse visa rather than the fiancee route.
So what I've learned is that courthouse marriages vary by county, so you'll just have to double check with where you're getting married. For us in Ohio we had to go to the courthouse where I am a resident of to get the license. We both had to be present, and I had to bring something that proved I lived there (i had lost my drivers license, oops, so I brought a credit card statement with my name and address on it and that sufficed). All my husband needed was his passport and they had him sign an affadavit stating he does not posses a social security card. Then on the same day, we took the license to the courthouse in a different county to have the ceremony performed. Then right after that we had to go back to the courthouse I'm a resident of to have it signed and certified. Sometimes you aren't able to get your certified copy that day and they mail it to you. So if you plan on applying for the spouse visa right after the wedding, double check you'll be able to get your certified copy the day of.
In Ohio, we did not need a witness, and birth certificates aren't required. The only thing is that if one of you has been divorced before, you'll need to bring the official divorce decree.
It sounds like you've done your research for weddings in NY though, so I'm sure you're good. Through the Columbus Municipal Court, they have a marriage hotline which I called to double check everything - I'm guessing NY might have something similar. It does seem way more straightforward than the whole visa process hahaa, and when you get to the courthouse they tell you exactly what needs to be done.. you can't really mess it up
Good luck, and congrats again!