1. The washer is beveled to allow for the slope of the tub so that the drain can come down straight. Just as you suspected. Usually that means the fat end is on the bottom but not always.
2. Tighten the strainer into the shoe. This is not true for a lavatory pop=up or a kitchen sink strainer. Putty is fine on a cast iron or steel but use silicone if you have fiberglass.