1)
You are correct. Suggest you get into a junior boarding school first and transfer to a high school later. This will give you more time to improve.
2)
If the junior school you are at has a high school division then you don't need to apply again. However, if the junior school you are at does not have a high school then you will need to apply to another school. At that time, for the top schools, you will need SSAT and TOEFL scores.
3)
For now your child will need to take TOEFL or TOEFL Junior test, SSAT (only for some schools and will be on their website). When you apply to high school, your child will need to take the SSAT again as well as the TOEFL.