Just to update this issue; it's been a while
...after I last wrote, I soon noticed one pretty definite trigger for the connection issue: sometimes I start the computer and the software, start working and then notice that my ethernet cable is plugged in (which gives me some hum, perhaps because my router is on another circuit? or because of the internet "signal" itself?). I then simply remove the cable (with everything still running), and that's when things go wrong. The problem seems to be fixable almost instantly by changing my computer's name (to whatever; it must be just the act of doing so). I haven't tried to replicate this intentionally, since the problem has cost me a lot of time and nerves and I'd rather not risk things not working for an indefinite period of time in the case that this _isn't_ the solution.
Anyway, it seems to make sense in the context of what others have written about such things, so perhaps mentioning the experience could be of value.