Thanks so much for everyone's comments here. I was able to use them to guide me through fixing a similar problem with my S50:
Canon S50 broken sliding lens cover / power / lens extension repair
My thanks to the following for references below getting me on the proper path. Refer to them for additional details, notes on other similar models, and photos.
Holding the camera as you would if you were to take a picture, the two screws on the outside of the case on the left side near the USB flap, as well as the one screw underneath the USB flab, only hold on the BACK of case. There is no need to remove these screws if you only want to remove the front panel, which is all you need to do if you are working on the door. In total, to remove the front panel, I had to remove 2 screws from the bottom on either side of the battery cover hinge, 2 screws from the bottom on the front side of the tripod mount, and the two screws on the right side (not the ones that are on the silver part). Then it's just a matter of gently prying the cover off - I used a tiny flathead screwdriver to assist. This should not take any significant effort - the case is indeed well fitted. Be careful, however, because...
A ribbon cable runs from a small watch-type battery (which is attached to the inside of the front panel behind the sliding door next to the lens) to the top of the inside of the camera next to the zoom switch. Here it attaches by sandwiching itself between white plastic parts to make contact. However, there is a brown "lock", shaped like a long rectangle, that slides in on top of this piece. After removeal this "lock" looks kind of like a jumper cable from a personal computer hard drive. It has to be pulled out first before you can pull out the ribbon cable. The ribbon cable is only just long enough to connect things, and definitely short enough to make repairs really annoying, so it needs to be removed to do any real work on the front panel. All this needs to be done carefully, though, so as not to damage the ribbon by too much flexing or scoring.
Inside the front cover, there is a rectangle-shaped piece of metal with three screws. I took off those screws carefully - underneath there is a spring on the top right that will pop out (I can't figure out what this does), and a spring that is attached to another metal plate underneath. This plate and it's spring slide horizontally (actually, from top to bottom of the camera), activating with the sliding door. I'm not sure what these are really for, though.
In my case, removing this horizontal spring resolved my problem. The metal plate it was hooked up to now slides freely, and it doesn't seem to be a problem getting it to slide like you would expect it to. But this doesn't seem to have been the only reason the problem went away. I was unable to get the cover off, so I could not get to anything between the sliding cover and the outside of the plastic case. After much jostling (with the cover off, though I probably didn't matter) two tiny half-moon-shaped gear pieces fell out. It's hard to tell for sure, but these were likely ONE fully round gear that had broken apart. Don't know whether this was the original problem or whether this was due to my jiggling. In any case, there is still something loose behind the sliding door, but I can't get in there to see it and didn't feely like prying it off. I couldn't get it to slide off past the closing point either, as one person suggested.
There is a much smaller small strip of metal underneath and perpendicular near the top of this plate. It makes contact with another gold-colored strip of metal, and when it does it pushes the gold piece which is clearly connected to the battery. You can tell that this is the key component that turns the power off and on (thus extending and retracting the lens).
If the battery cover has been removed, it must be reinstalled before the front cover can be installed. Also, annoyingly, the camera could not be turned on (and thus the repair could not be tested) until I screwed the front cover back on (though I did not test it screw by screw). There is a flat metal plate that attaches to the top left side of the inside of the front cover and slides inbetween other parts during assembly. It looks like it is meant to be connected by two rivets, however one of my rivets came off (don't know if I did that or not), and the other eventually did later with my jostling. It is roughly triangular in shape, with various holes and slide spots. When I reassembled the camera, I did it without this part and everything seems fine.