Is this coat actually made of wool?

No. The listed material is polyester, so the "woolen" in the name describes the look, not the fiber. Expect a synthetic coat, not a wool one.

What sizes does it come in?

The size run goes from S up to 8XL, which is wider than most overcoats. Check the measurements for your size rather than assuming your usual letter size fits.

Is it warm enough for winter?

Probably not for cold winters. Polyester of unknown weight gives limited insulation, so it suits fall, spring or mild climates, or layering over a sweater.

How should I care for it?

Follow the care label. Polyester generally tolerates machine washing on gentle and low heat, but a structured coat often keeps its shape better with spot cleaning or dry cleaning.