It depends on whether anything vents into it--a water heater, furnace, etc. If nothing does, then you can run a duct from the range to the chimney--my kitchen's chimney already has a hole for a vent which was used by the old stove--now I only use it to add a draft during very hot summer days, the rest of the time it is covered by a brass cover. That same chimney has another hole in the bath above where an old heater vented.

It was common practice for a house to use the chimney both for heating and cooking ventilation from the old wood stoves. My chimneys were put up in '08, and are lined with flue tiles, so you may find yours the same, with no need to re-line them if an inspection shows it is intact.

It should be surprisingly easy to remove the chimney yourself if you want to do it--start at the top of course and work your way down using a heavy hammer to loosen the mortar joints, then you can reuse the bricks elsewhere; you will then have to patch the holes in the floors and roof, but you could also use those as chases for wiring.

Running a stove hood exhaust up 30 feet to the roof will be very ineffective. Simpler to run it down and out the basement wall. That's done all the time for downdraft island stove tops.

To clarify: if you use a downdraft exhaust it must be with cooking equipment that is designed for downdraft.