Are Shimano 9-speed derailleurs backwards compatible with 8-speed cassettes and shifters? Or is the cable pull different between the two systems? i.e., can I buy a new 9-speed Shimano RD and have it index properly with my existing 8-speed cassette and shifters?

And while I'm at it, what new medium-cage derailleurs will handle a 32T large cog?

9 speed will index fine on 8 speed. Check the Shimano web site for the level of component you are looking at to see the max cogs size for the various RDs.

as long as you avoid dura ace and shadow derailers, the compatibility is fine.

There are two different medium cages. One is MTB, the other is road.
The road is specified to 27T max, but usually 30T is okay.
The MTB is specified to 34T max.

Thanks, I figured most mtb derailleurs are equipped to handle larger cogs. The reason I need a medium cage that can handle a large cog is because this is for a tandem with 20" wheels, so a long cage would rub the tire (not to mention risk getting eaten by rocks and curbs). But I need mtb gearing to negotiate some serious hills and heavy loads.

Friends who have trikes ride with 20 and 16 inch wheels and mountain derailuers with no problems.

What size tires are they running? I bolted up a long cage Deore LX, and found that the cage rubbed against the 20x1.9" tire when in the largest cog. A narrower tire may allow it to clear, but then I don't want a narrower tire on a tandem that sometimes weighs over 400lb loaded up.

If you can't use a MTB derailer I've been able to get to at 32T with a short cage road derailer, provided a long enough b-tension screw. It's not ideal but it might be your only option for running those big cogs.

Tandem with 20" wheels? Don't think I've ever seen anything like that before. Mind posting a picture?