The university that I would like to attend requires me to be at least 21 years of age and take the SAT I, which means that I have two more years to take the test before I apply. Will I be able to take the SAT I during those two years? Also, will I stand a better chance of being accepted to universities if I take the SAT II as well? Thanks!
Yes, you can take the SAT after finishing high school. You're actually in a better situation than most students because you have extra time to prepare for the test. Just select "I am no longer in high school" when the College Board asks for the name of your high school in the registration process. When you have to choose a test center, pick whichever option happens to be closest to where you live. For more information on taking the SAT as an older student, check out this article.
As for your question about the SAT II, it depends on where you apply. Some schools don't care about the SAT II, and some highly recommend or require that you take two or three subject tests. You can go here for a list of the policies of colleges that require SAT IIs. We've also written an article that will help you decide which SAT IIs to take based on your interests and goals.