Well, of course, I don't know its not on. That's why I'm here asking questions. What I do know is that it isn't working as the limited documentation I can find (much of it not from apple) suggests it should. Sorry if I sound terse myself, but if Apple is going to release software that is supposed to make it easy to set things like this up, and make advertising claims like:
"OS X Server is perfect for a studio, business, hobbyist, or school. It’s so easy to set up, who needs an IT department?" http://www.apple.com/osx/server/
and:
"Configuring OS X Server is almost as easy as configuring a desktop computer. Simply launch the Server app. An assistant walks you through the initial setup and helps verify network settings and define your administrator account. Next add users and groups and enable the network services you want to deliver to your organization." http://movies.apple.com/media/us/osx/2012/server/docs/OSXServer_Product_Overview .pdf
...then they are going to have to expect frustrations like mine.
Now, back to my problem (assuming you've made it this far): server.app says DNS is on. I've posted various things about what the configuration page on server.app says (not on this post, but my other one specifically about trying to get DNS to work). However, when I set an outside machine (not behind my server's firewall) to use my server (the one that server.app says DNS is running on) for DNS, it hangs and doesn't "work". I'm sure something may be working, but it isn't working as Apple's advertisements (and server.app itself) suggests. So it isn't "disingenious" for me to blame the tools. It literally says its on. It also isn't responding to clients, and there is nothing in the setup that would lead a reasonable person to expect that it shouldn't. Better documentation from Apple would help. Did you say you knew where that might be? I see this: https://help.apple.com/advancedserveradmin/mac/10.7/ but that is for 10.7, not 10.8. This is part of my frustration.
My guess is that OS X Server is not configured to allow outside (not behind the server's own firewall) queries about DNS. This, I'm sorry to say to all you much-more-IT-savy people than I, is not at all obvious to someone like myself, hence my questions here to you all. If there is some configuration file I need to edit, outside of server.app, by all means, tell me about it.
Finally, regarding your comment to check the logs, I will certainly do this when I get time (this isn't what I get paid to do, so I'm working on this while I can). However, I fully expect that I won't know what I'm looking at, or how to interpret it. This is why I've been trying to get some intuitive understanding of what is likely going on first, because staring at log messages tends to be unproductive if you don't really know what you are looking for.
In any case, I do appreciate those who have tried to give me hints about what might be the problem, as I work through all this. I'm sure you can understand the frustration of someone who put faith in Apple's advertising that it would be easy as pie.