Ron -- Thanks for that! That helps and explains a lot. So I know THAT part is working fine.
Here's what I've done.
I have 5 sites. I imported the data into a Wordpress 4 MU environment. I set up rough designs for each site. At this point I could access dashboard and website for each site.
Two were ready to go -- the main one and one other. I enabled the Domain mapping plugin and mapped the first 2 sites. Both of those sites work. I can get to the dashboard for the main site and I can get to the Network dashboard from the main site. (I'd missed that limitation.)
When I try to access the dashboard for the #2 site, I get the dashboard for the #1 site.
When I try to access the dashboard for the #3-#5 sites (unmapped) I also get the dashboard for the #1 site.
When I try to access the unmapped sites via the dashboard, I get a "page not found" site on the #1 site.
It is entirely possible that I've messed up the site. At one point, my #1 site was actually #3. I've followed directions elsewhere for changing this and it seemed to work really well before I began domain mapping. So THAT may be part of the problem.
Also, I initially had this set up in a subdomain and then read that for domain mapping to work, I had to move to the root. So I moved it following the directions -- but again the possibility for error is high. I tested the sites with domain mapping turned off and they are all where they are supposed to be, all working, all dashboards going to their individual dashboards.
If I could get #2's dashboard to go to #2's dashboard (as opposed to the main site), I'd be comfortable that this was going to work.
If I had something definitive that says "it works with 4.0" then I'd go ahead and check to see if I've got an error in my database.
I'm aware that I may need to rebuild the whole thing correctly (avoiding the mistakes I made during this process). I just need some assurance that it DOES work with version 4.0 before I do that work again.