I do not own a skill derby, but have worked on lots of bingos. Those are the same leather clutch washers used in the bingos. Bally had at least two different diameter motor shafts, and separate clutches for each. NOS clutches appear on ebay pretty frequently.

It is important to note which side is the slick and which is the rough side when disassembling/reassembling. The rough side is usually the side that engages the thing that should be turning. Clutch washers should be soaked in neatsfoot compound. I will take a set of washers, and if they are not misshapen or damaged, will clean them with alcohol, rinse them quickly, then soak them in neatsfoot compound for about 20 mins. Once they are soaked, I take them and pat dry on a paper towel. They should be lightly greasy but not dripping.

I know this doesn't answer all of your questions, but new clutch washers make a lot of difference in a mechanism like this. Hope this helps a bit. You will have all kinds of mechanical problems if the leather washer is worn (with smooth rotation, especially).

I have never worked on one of the games with the horse units, and without a photo of what you mean I'm not sure I follow. I'm sorry.

If it has been worn smooth, you can try roughing it up (I'm not sure that would work) or just replace them.

Sorry I'm not more help, hopefully someone chimes in with experience with this mech. @steveintexas?

That was probably the amazing baldtwit 's site.