Ok, never mind -- I think I found the answer in someone else's post -- looks like I have to delete all of these songs from my iTunes library, but leave them in the cloud -- and that will get me the download icon and let me play them. Really counterintuitive, and now I'm wondering what happens when I sync my iPod with iTunes -- won't the deleted songs also get deleted from my iPod, making me have to redownload them? But I'm sure I'll figure it out.
Instead of deleting all the songs from your iTunes library just create a new, empty library and enable iTM. This will allow you to then stream the music in iTunes without messing wtih the original, master library.
As to your question about syncing, with iTM enabled on iOS devices direct syncing with iTunes via USB or Wi Fi is disabled. All music is managed directly on the iDevice.
I agree with Michael. What's happening is because you broke the link to your external drive. As far as Match is concerned, you still have all of those songs available to you locally.
If you've never created a new library in iTunes, it's easy and painless. Switching back to your main library after you hook back up your external drive is easy too. On a Mac you just hold down the option key when launching iTunes to get the prompt to either create a new library or to link to one you already have - simple as that.
For what it's worth, I set up a second library after enabling iTunes Match. All I put in it is new music I want added to the cloud. It saves me a lot of time and if I really want a copy of the songs locally in my main library, they still show up via Match as downloads, so I can grab them anytime.