What is the point of non-commutative probability?
In a very tiny, very densely packed nutshell, here's a partial answer to your question 1 (as far as I understand it, which is not very far at all; and I really cannot say much of anything about your questions 2 and 3).
Regarding what it means for probability to be "commutative", it's not just that $E(XY)=E(YX)$, it's that $XY=YX$. In other words, multiplication of random variables is a commutative operation on the set of random variables. This is supposed to be "obviously true", because random variables are real valued, and multiplication of real numbers is commutative.
Regarding the meaning of "noncommutative probability", the first idea is to think of "commutative probability" as being a theory of the commutative algebra of random variables. In other words, one focusses not on the state space, nor on the events in the state space, but instead one focusses solely on the abstract set of random variables, and on the operations of addition and multiplication and scalar multiplication on this set. A set with two binary operations $+$ and $\times$ and with a scalar multiplication operation, and satisfying various axioms that govern the behavior of those operations (in particular the commutativity of multiplication), is called an "algebra" or more precisely a "commutative algebra".
One then makes a vast leap from commutative algebra into noncommutative algebra, to see what properties of "commutative algebras of random variables" can be generalized in the setting of noncommutative algebras.
What's going on here is that one conveniently "forgets" where random variables came from, in particular one forgets that they are real valued functions on measure spaces. One then proceeds to formally generalize the properties of commutative algebras of random variables to the setting of noncommutative algebras. Then one attempts to "remember" where random variables came from in this more abstract setting. That confusing but enticing statement you point out is an example of the "forgetting-remembering" process. This turns out magically to be a very fruitful process and results in some very interesting and important mathematics.
In short,
Multiplication of random variables is commutative in standard probability theory but not in noncommutative probability.
Terence Tao uses noncommutative probability to understand random matrices; I wrote the blog post you linked to in order to understand quantum mechanics, and in particular quantum probability, which is noncommutative.
The uncertainty principle. This is explained in my blog post that you linked to.