So after two full days of searching for a reason for all the weirdness, they found it: The timing belt pulley inside the engine was loose, and was causing timing problems as well as broke off a (small) piece of the crankshaft.
As I type this, the pulley is not coming off the crankshaft, because the piece that was supposed to hold it in place on the crankshaft, the woodruff key, is rusted onto it, and won't let the pulley come off.
So my options now are, if the pulley comes off, get the few relatively inexpensive parts replaced, and get the missing piece in the crankshaft filled in with JB Weld, and pay for labor; if there is no way to get the pulley off, it will be cheaper to install a rebuilt engine.
The folks at the shop are really nice to me, so they will charge me a flat fee for labor in the first case. The second case, well, I try not to think too much ahead. I'll deal with it if and when I have to.
My car's "attending physician" picked me up at work to show me the damage. It was really sad to see Bonnie on the lift, with rags covering the valves and other parts they were not working on, with just the front part of the engine exposed, it looked almost like a body on an operating table.