I am thinking of putting a slide-in range with downdraft venting in an island in the center of the kitchen. Do I have to vent the downdraft system all the way to the exterior of the house? Can I have it vent to the space in between the floor and ceiling below? Or, do you have a better suggestion?

You have to vent to the exterior of the house.
The reasons being
1. If the bay (the space between the floor joist) is sealed you won't get any air
in. You can't put air in unless there is somewhere for air to go out.
2. If the bay is not sealed you will be putting in greasy fumes and odors to be
Carried all through out the house. The purpose of the vent is to carry fumes and
Odors out of the house.

Open the ceiling below and use flexible duct to vent to the exterior.