Types of lathes, necessary accessories, etc., have been discussed earlier in this thread, Evan. I’ll try to summarize a bit, but I would recommend reading through it when you get some time.
In general, a good tabletop lathe would be sufficient for turnings 8 inches round or so and smaller. Spindles would be 12 inches max or so. Forget the baseball bat for sure. For shaving brushes, bowls, etc, it would be more than fine. Expect to pay about $500 US for a good one. High Speed turning tools are expensive. However, with your metalworking ability, you could make some scrapers and a parting tool very easily. On this thread,
@9nein9 detailed making nice turned handles for tools as well.
If you want, you can get away without any special chucks, but I am a huge fan of a 4 jawed chuck, as well as a collet chuck to hold a brush by the hole. Without a chuck, you will be drilling and finishing off of the lathe. There was a discussion of chucks on here as well.
As for a collaboration, I’d be game!
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