-Is it correct that non- U.S. Citizen can recieve 1 introductory flight but to further their training or get on a 2nd flight they need to be cleared by TSA? Or any training even the first one needs to be approved by TSA before it happens?

-As a person with JAA private pilot license, if a person was to visit U.S. for a week and get a flight lesson with FAA CFI, can he/she log that time or does that person need some sort of FAA approval to log that time?

Thanks!