I set up Thunderbird for one of the accounts, and that worked.
However, right after I used Thunderbird, I checked the webmail again and that was "working" too. So I've no way of knowing whether Thunderbird would have worked if webmail definitely hadn't been.
By webmail "working," I mean I first got the gibberish mentioned at the beginning of this thread and had to try to log in a second time, but I did get in then.
But the webmail was slower than molasses in January. Which I suppose means the problems (of the gibberish with need to log in twice and not being able to log in at all) are due to some sort of server overload or malfunction at Dreamhost.
I doubt it was something wrong with my connection to Dreamhost, because Thunderbird worked very quickly.
"See, Hear, Smell, Taste, Touch, Touch, ... Yes,