I'm back again. Here's what happened. I could open the DVD with the emergency open, using a pin. It didn't read any CDs or DVDs I put in it, though. I tried uninstalling and reinstalling it. Didn't make any difference. Windows sees the DVD player as being there and says that it's working properly.
Now the new stuff. I bought a used CD-ROM because there was software I desperately needed to load onto my computer. So I make the CD the master, the DVD the slave and reseat both in the computer. With that done, the DVD starts working again! It worked for several days and then stopped working again, won't open on its own, etc. From the time it was working the second time to when it stopped working I didn't install any new software and didn't do any Windows Updates.
I'm beginning to think that maybe a connection isn't working somehow, that something's becoming loose. Neither the CD nor the DVD has been screwed back into the frame yet because I was waiting to see what happened with them, but the DVD was working just fine for several days and the computer hasn't been moved around hardly at all. (Except for attempting to hook up a scanner.)
Oh, wait. I wasn't around when a friend was trying to hook up the scanner. Didn't have a driver for the scanner though, so no software for it had been added to the computer.