FWIW, I think I figured out what the problem with my monitor was.  It's a wide-gamut monitor, as many are these days.  That's why the reds were so eye-searing.  Windows doesn't get along with wide-gamut monitors.

Apparently, you are supposed to do a hardware calibration of a wide-gamut monitor, then use that profile in all the programs that will take it.

I did that, and things are a lot better now.

And if anyone cares, I went with the Spyder4Pro.  I decided I didn't need the most expensive version, because I don't have a two-headed system.  But I needed more than a SpyderExpress, because I want to use it on more than one computer.