hey MelanieT,
Just try to be relax. I think you are overwhelming your self.
1. If you are doing adjust of status here in america you should pay your fee in Cashiers check or Money order. here is the instructions for sending the money:
http://www.travel.state.gov/visa/visa_4763.html
Paying in cash is not an option for you. People who do consular process can pay for their visa in cash.
2. "Paper work" is the medical form I-693. If you have visited a USCIS approved civil surgeon and you already got a sealed envelope, you should be fine.
3. There is no restriction for traveling inside the US. You can also travel abroad (if you have a valid visa with multiple entry permission) before you submit your application to the USCIS. You if you are in the US and planning to travel to any state, you are fine.
As far as being eligible for AOS, If you have been on your visa for the whole time, without working outside campus and other legal issues, you are eligible to do AOS. you don't need any letter for that.