Does the DCS361B come with a battery?

No. It is sold as a tool only, so you need a 20V MAX battery and charger. The listing text mentions a charger and battery, but the tool-only model number means they are not included.

How big a board will it cut?

It cuts up to 8 inches at 90 degrees and handles 3-5/8 inches of nested crown and 3-1/2 inches of base vertically. Wide stock beyond that needs a bigger saw.

Can it bevel both ways?

No. It is a single-bevel saw, so you tilt it one direction and flip the workpiece for the other side.

Is it good for trim work?

Yes. The 11 miter stops, cam lock handle and cut line system suit baseboard, casing and crown. The 7-1/4-inch blade is the main limit on large profiles.

How does it compare with a corded 10-inch miter saw?

A corded 10-inch saw cuts wider stock and bevels both ways, but it needs an outlet and weighs more. The DCS361B wins on portability and setup speed for trim jobs.