I'm assuming I might only need to fly a few checkrides with a CFI at the school, and spend some ground-time for the navigation exercises, apart from actually flying the nav-exes.
You assume incorrectly.
You will a FAA Medical and you must complete the FAA written exam. The FAA Checkride also consists of ane indepth oral exam which you have to pass before completing the actual flight portion.
The thread provides all the information that you need on what the visa requirements are. If you are still confused - best read it again and read the attachment that I provided in my previous post.