Q.If you look at the sky in a particular direction through a polaroid and rotate the polaroid around that direction, what will you see?

Since light is a transverse wave, its electric field can point in any direction within the plane perpendicular to its direction of travel. Ordinary light from a source is UNPOLARIZED -- a random mixture of many independent wave trains, each with its own random field direction in that plane. A POLARIZER (e.g. a Polaroid sheet) transmits only the field component along one fixed direction (its polarizing axis), producing PLANE POLARIZED light with a single, well-defined oscillation direction; the plane containing the propagation direction and the surviving field is the plane of vibration, and the plane perpendicular to that (but still containing the ray) is the plane of polarization.

Because an unpolarized beam has its field spread with equal likelihood over every angle, averaging always halves the intensity through a first polarizer, REGARDLESS of that polarizer's orientation. Only a SECOND polarizer's orientation (relative to the first) matters -- governed by Malus' Law -- letting light be smoothly dimmed from full intensity (parallel axes) down to zero (crossed axes, 90° apart).