Why the direction of the omega (angular velocity vector) is along the axis of rotation? Also for angular acceleration

We can define the angle as the area sweep by a vector in the rotating plane with its strating point at the rotating position. As the vector rotates a angle $Delta theta$, the area swept by the vector is: $$ Delta A = r^2 Delta theta $$ Therefore, we may define the angle as the area swept divided by $r^2$. The area is a vector quantity with direction on the normal direction of the area. This defines the direction of the angle $omega$. I will show that the definition of area is more general that the "angle" itself.

Case 1: Consider that the rotating vector is not ristricted in the plane, e.g. it rotates in a corrugated surface. The resultant rotation angle $theta$ won't be the sum of the infinitismal angle $Delta theta$ $$ theta neq sum_i Delta theta_i $$ But it is equal to the sum of the infinitesmal area. $$ theta text{ } hat{theta} = sum_i frac{Delta vec{A}_i}{r^2} $$ The $hat{theta}$ denotes the result vector direction of the vector summation.

Case 2: A solid angle $Omega$ can only be defined using area $$ Omega = int frac{hat{r} cdot dvec{A}}{r^2} $$

Hope this helps.