A bit of bad news... As I was removing the chrome trim from around the windshield, an inverted L-shape crack appeared in the windshield. It's about 6" on each leg of the L, so the windshield is likely toast. I did a quick Internet search and new windshields aren't too expensive. I was going to remove the windshield for the new paint anyway, so this just reinforced my decision. I won't be doing any serious driving so I'm not worried about a catastrophic failure.
Now on to the mystery spots. First, the "before" pictures. Here's the driver's side: