I'm in your position! Kind of - I'm in year 12, applying this fall, but I first became interested in US study a year ago so I've been pretty much researching it since then!
I haven't sat my SATs yet but you'll need to do them soon - I believe it's too late for you to register for the May date, but you could potentially register for the June date. However, less schools offer the June date as it's in the middle of exam period. After the summer you'll have the October and November dates to choose from. Honestly, if you're doing SAT IIs in the subjects that pure studying for AS I would do the SAT IIs instead of the reasoning test in June, purely because a lot of the information is relevant in both AS and the SATs- I'm speaking from personal experience with Biology and French. Then, you could sit the Reasoning Test in October.
The application process for a lot of private universities eg. Columbia is very extensive, are you aware of the use of the Common App? The more colleges you apply to, the more essays you're going to have to write. I'm applying to 9 and also 1 in Canada, so I'll probably start during the summer, as well as doing my personal statement.
Have you looked at liberal arts colleges yet? There are some that have higher acceptance rates and actually offer international financial aid such as Skidmore College, which offers a few internationals a need-based financial aid award each year.