If you have a choice, then your child should go to a boarding school since this will give him/her more opportunities during the day to learn English. If your child is at home with you then he/she will not have as many opportunities speaking in English.
Socializing with friends is a way for a single child to learn life skills, such as sharing, cooperating, caring about each other and taking care of each other. You can always stay close by the school.
All the top schools would like to see SSAT and TOEFL scores.
Application requirements are different for every school so you need to check their websites.
If you don't have a green card, that will only affect your application when you are applying to a public school or if you are applying for financial aid.
Yes, all the top schools require US students to also take the SSAT as part of the application.