Thanks for the tips!
After a lot of troubleshooting and endless hours rendering, I've narrowed down the causes of a lot of these problems.
Too-bright fur: Lighting and shadowing, obviously. I needed to add Fur Shadowing attributes to all my lights, and adjust the self shade and self shade darkness sliders until the fur wasn't too bright or too dark. Using 32 GI_Joe lights, I ended up with about .3 self shade and .8 self shade darkness. YMMV. I also played around a lot with the specular settings, there are lots of them on the fur attributes and on the lights. I'm still not thrilled with the lighting on the fur, but it's not nearly so distracting.
Gap between fur and skin: D'oh! It took me forever to realize this was a problem with my UVs. Fur is unbelievably sensitive about UV mapping, and if you have ANY overlapping or pinching at all, bad stuff happens. I'm slowly going through and tweaking my UVs, and the gaps are going away.
Patches of baldness: Again, UV overlapping. Currently, the only baldness I have left is on the cat's shoulders, where I still have some overlapping faces. When I fix those, I expect fur to grow normally there.
Whew! This has been quite an exercise for me. Especially given I just started learning UV unwrapping a few days ago.