TB 3.1.2 will not install any extensions or add-ons

(Win7 Home Premium 64-bit)

Thunderbird checks automatically for updates to add-ons, which is great, but recently when I clicked the button to install updates, it hung. It should have downloaded and installed 7 updates but it opened a blank screen (the Updates screen) and just sat there. When I clicked the "Show information" button, it only displayed information for one add-on, and I couldn't tell which one it was because the list pane remained blank. (I let it sit for eight hours once while I was out and about, no change when I got back.) To open TB I had to cancel the updates, but when I'd do that it warned me that one download would be canceled (presumably the download for whichever add-on the information was about).

I thought I was able to install the updates one by one, by de-selecting the checkbox for all but one at a time. When I clicked install, the item would disappear from the updates list so I thought they were installing -- but no, all the add-ons are still the old versions (which I prove by choosing Check for Updates in the tools menu).

I have since found out that I cannot install ANY .jar or .xpi file, either by using the Install button from the Themes or Extensions screen and selecting the file to install, or dragging and dropping the file into the Extensions or Themes window. I've tried this with several different files.

All the extensions I currently have were installed in a previous version of Thunderbird. There are other things in 3.1.2 that don't work either (links in the Links menu, for example, or links in the Add-ons window) but used to work in earlier versions.

This is highly aggravating! I need TB to function correctly. How can we find out what's going on?