Thanks Timbo and Marvo for the replies. I appreciate you confirming I'm not going crazy. Here's a few quick follow ups/thoughts:
1. The voltages are correct in the picture. I quadruple checked everything because it didn't make sense to me (and I've done a lot of wiring for my job, although in a laboratory setting where you start and finish the job yourself).
2. The 'White' wire is actually a black wire, with white tape on it.
3. When I tried I assumed the lights wouldn't work since I never get a differential between the two wires in either switch state, but I know the previous homeowner had a light here so I went for it, but it didn't work obviously. The only thing I can think of is if they had one light fixture wire connected to the black (the one that actually switches on and off) and then the other wire connected to ground. However, then my ground would be live when the lights are on which sounds like a BAD idea.
4. You guys confirmed what I was thinking, which is the previous owners had no clue what they were doing and my house has screwy wiring. This isn't the first thing I've run across that makes no sense to me. For example, a switch I've changed out in the past seemed to be missing a neutral at the junction box also. Did old houses just not use a neutral?
5. Last thing I might check is just to make sure a neutral didn't migrate its way back out of that junction box somehow.
6. I plan on a kitchen remodel in a year or two so may just deal without a light over the sink until then. We plan on a complete remodel so we may need an electrician in then anyways.
