do eigenvectors correspond to direction of maximum scaling?

There is something called the matrix norm, $||A||$ which according to Linear Algebra and it's Applications by Strang is defined on $\mathbb R^{n\times n}$ to be $$||A||=\max_{x\neq0}\frac{||Ax||}{||x||},$$ where $||x||=\sqrt{x^Tx}$. From this, $||Ax||\le||A||||x||$. So the norm measures the "amplifying effect" of $A$ on a vector $x$. Equality happens at the optimal $x$. The value $||Ax||/||x||$ is always nonnegative. We square this to get $$||A||^2=\max_{x\neq0}\frac{x^TA^TAx}{x^Tx}.$$ Matrix $A^TA$ is symmetric. This is known as the Rayleigh quotient, as it is well known that its maximum is attained when $x$ is the eigenvector corresponding to the largest eigenvalue. Then the maximum happens when $x$ is the eigenvector of $A^TA$ corresponding to the largest eigenvalue of that matrix.

No - An eigenvector is the "input vector" of a matrix that "outputs" a vector in the same direction. If you really want an intuitive understanding of Eigenvalues and Eigenvectors, follow the link. (I don't have the reputation needed to post images.)

Linear Transformations & Eigenvectors Explained