I have solved the problem with Dropbox. It was the proxy.
On my MacBook, Dropbox uses the system settings, and the proxy is set in there, so no further settings need to be done. But on my Linux desktop, I have no system settings, so it was not using the proxy.
So I needed to get Dropbox to use the proxy. I had the same problem with slackpkg and sbopkg and solved it by using the HTTP_PROXY environment variable. That works fine for slackpkg and sbopkg. So I set it up to work the same in my /etc/rc.d/rc.dropbox and tried it out. It didn't work. I tried a few other things like running Dropbox from a terminal with the environment variable set in different ways, but could not make it work. I did some more reading on the Dropbox wiki, and it seems that the easiest way (if you are used to using GUIs) is to hit the little Dropbox icon and enter the proxy settings manually.
I have a problem with that. I don't run a desktop like KDE or XFCE. I use a window manager - fvwm. So I don't get a nice little system icon that shows that Dropbox is running and can control the preferences. I looked for a separate program that I could run, but it appears that the Dropbox daemon is also the setup dialog box and visual indicator. Can't have one without the other.
I shut down fvwm and started up again in xfce. Ah, there's the little Dropbox icon (just like on my Mac, except at the bottom) and I can right click on it, select Preferences, and set the proxy. The first time I set the proxy and saved, it came back and told me there was an error with my settings and dumped me back in proxy edit window. I looked at the my typing, it was all okay, so I saved it again. This time it didn't complain. I then went and did my tests again, and now it updates on my work Linux desktop when files are put in on my other boxes. So it's all working now while I am running xfce. I dropped out of xfce, rebooted (just for the hell of it) and started again in fvwm. Dropbox continues to work. It continues to use the proxy settings entered in the Dropbox GUI configuration.
That will do for now.
But this brings up the point that I should be able to set a system wide proxy setting on this desktop. I am going to research and see how this is possible in Slackware.
And I want to know if these notice-icons can be displayed and be functional under FVWM. More research needed.