applying and studying at university

Using your example of Harvard:

Harvard takes 2000 first year students every year. 98% of them stay, so there are very few places for transfer students to come in. Last year they accepted 13 transfer students (out of 1,432 applications).

Yes

That’s what I was saying above: you apply to Harvard as a university, not for a specific department. As a transfer student they will expect you to have good, specific reasons for transferring to Harvard. Not having any idea what you want to study will make it harder to write the 'why I should be at Harvard" essay.

Also, entry to a subject (called a major) is NOT based on SAT subject tests (which are for secondary school students anyway).

No.

No.

There are Education USA offices all over the world- @Gumbymom was giving you the Moscow link b/c you mentioned speaking Russian & she was being helpful.