"E[foo]" is syntax to mean the expected value of the random variable foo, roughly speaking the mean value. (For instance the expected value of a dice roll is 3.5. The terminology is slightly suboptimal, since we will never expect a dice to come up 3.5.)

Hence the "E" itself is called an "operator". It can be applied to a random value in order to yield its expected value. You can read up on it here: https://en.wikipedia.org/wiki/Expected_value

The definition "E[x] = mu" is correct, though I would write it the other way, as "mu = E[x]", as it's the variable mu which is being defined.

The v's disappear because of a suppresed calculation:

sigma^2 = E[ (v^T x - E[v^T x])^2 ] = E[ (v^T x - E[v^T x]) (v^T x - E[v^T x]) ] = E[ v^T x v^T x - 2 v^T x E[v^T x] + E[v^T x] E[v^T x] ] = E[ v^T x x^T v - 2 v^T x v^T E[x] + v^T E[x] v^T E[x] ] = E[ v^T x x^T v - 2 v^T x E[x]^T v + v^T E[x] E[x]^T v ] = E[ v^T (x x^T - 2 x E[x]^T + E[x] E[x]^T) v ] = v^T E[ x x^T - 2 x E[x]^T + E[x] E[x]^T ] v = v^T E[ (x - E[x]) (x - E[x])^T ] v = v^T E[ (x - mu) (x - mu)^T ] v = v^T Sigma v.