what are the dimensions?

Dimensions are 36" long x 20"wide by 9.5" tall. All in inches. Roll out easy under bed. Make sure you have at least 10 inches of clearance under bed. Stain nice right out of box no need to sand. 2 coats better than one.

Are the wheels such that the drawers could be wheeled sideways? I would want to orient them 90 degrees to what is shown.

The wheels are one directional. They roll the drawer forward and back on the long side. However, if you had the tools, they could be turned to pull the opposite way. It would take such things as a router, a jig, and other hand tools. It could be done, but not easily. The drawer is actually designed to only roll one way. If you installed other wheels, such as casters, the drawers would no longer work under the bed since they would be too tall.

Does the 10 high inches include the wheels?

The wheels left the box about 1/4" off the floor. The 10 inches is the front face board which does not go under the bed but stops at the bed frame keeping the box from disappearing under the bed. The other three sides to the box are about 8" high. I had to lift the bed about 1" to get the boxes to clear the frame. I did this by changing to larger wheels on the frame feet. This box glides on carpet with ease and is larger than an any plastic box I've tried in the past. The construction is superior.