OK, it worked this time, after re-booting. Thanks Stephen.
I used the computer, wired to the router (with the wireless shut-off my laptop), for about 10 or 15 minutes, and didn't have any problems at all. I should probably use it a bit longer, though, just to be sure.
I disconnected it from the router and turned on the wireless switch to see how it would act, and it worked fine until the laptop completely shut off a few minutes later. I thought it might be the battery, since my battery is ~2.5 years old; so I plugged it into the laptop adapter and turned the computer back on. The battery icon showed it as plugged in with a 97% charge. This (generic?) battery has been great, btw - it replaced the original (not-as-good) Toshiba battery, and has been A-1 all this time. So, the reason my laptop turned off may not be the battery? I don't mean to bring another problem into this, but it just makes me wonder if this has anything to do with it. And just to clarify, I had unplugged the laptop from the adapter, brought it to another room where the router is, and used it battery-powered, to do the experiment with being plugged into the router. That is when the laptop shut off, after about 15 minutes, or less.
Another thing, after I plugged my laptop back into the adapter, I browsed for maybe 5 minutes to see if it would hang, and it only did so one time. The page eventually opened after a minute or so, though. A 'Report Problems and Solutions to Windows' message popped up (following the hanging incident I just mentioned), so I clicked on it. This was the reported problem for this incident:
I'm beginning to think maybe there isn't anything drastically wrong...