The answers to all of your questions are already on the board but let's give it a go:
You DO NOT need a visa for time building unless:
- A previous visa application has been denied
- You have a felony conviction, includes drunk driving
- You are a citizen of a country that does not fall under the visa waiver program (google it if you need to)
Under ICAO agreements member states will recognize/validate each others certificates/licenses.
The FAA process is free of charge.....
Read all about it on the FAA website:
Airmen Certification - Verify the Authenticity of a Foreign License, Rating, or Medical Certification
This process can take up to 60 days so don't schedule too tight.
Some more unsolicited advise, DO NOT pay in advance for a large chunk of time building without having seen the airplanes in question.
You are NOT limited to JAA certified schools in teh USA, it's a complete myth that you need to build "JAA hrs".
You are free to go to any school that bothers to answer your emails and that has safe aircraft.