If people from other countries apparently think Americans are "stupid and lazy" then why are most top universities in America?

Yes, like Cambridge and Oxford? They get the same education as anywhere else. I got a good education from Tempe for my undergrad and Holy Family for my Master's. Just as good as Harvard, except it doesn't say Harvard on my resume.

That said, it's just jealousy. We wouldn't have an immigration problem, illegal or otherwise, if the world thought so poorly of us.

It's called "cognitive dissonance" - when you believe two opposite things at the same time and somehow reconcile it in your mind. That being said, a good deal of the TV programming exported from the US (that US audiences don't even like - they make more money elsewhere) enforce that view - "everybody in the US is like the family on 'Married... with Children'".

No, but the impression lingers. This is because there are a large number of functionally illiterate people in the country. Our overall literacy rate is lower than Germany's, Russia's or the French. Jay Leno has amply demonstrated the fact that people he interviews at random on the street can't find Canada on a map. You might not think this is shameful, but I do. Most people in Europe speak 2-3 languages. That's not the case here. Our math and science scores for high school grads are another embarrassment.
Have you ever seen "A Fish Called Wanda"? Kevin Kline portrays the obnoxious American very well.

A good many Americans are stupid and lazy. They don't all attend those universities. You can find very smart and motivated Americans, but there are others who think they're entitled to a free living, even if they don't earn it.