That's how you know you're getting better - when the rules of logic cease to be optional and start becoming your default mode of thought. And if you think about it, isn't that how it's supposed to be anyway?
Quite the understatement. Harvard/Yale/Stanford don't do merit-aid scholarships so you would have to be super poor to get some financial aid and even then isn't likely to be a full-ride.
Doesn't Stanford offer free tuition for students of parents making "less than $125,000 annually"? (previously $100,000). However, I'm not sure if this only applies to undergrads. Just something to check out.