The clause is meant to inform you, the developer, that it is your responsibility (obligation to use that terminology) to ensure your application runs on the supported versions of Ubuntu, not Canonical's. You are not forced to test on all versions, but it is in your best interest to ensure it runs on the latest Ubuntu release and prior versions.
For the purposes of the software center and paid applications, you should test 11.10, 11.04 and 10.10. You always have the option of only supporting the latest version if you wish and just need to inform the reviewer of that requirement when submitting your application.
In the case of ARB apps (FLOSS apps) you will want to ensure compatibility with all supported releases.
I hope that helps clarify the terms a bit better.
On the page you link to it says:
Canonical provides this packaging service to you free of charge.
and goes on to explain how to lay out the tarball containing your binaries.
If you create such a tarball and submit it via MyApps then all the packaging will be done for you. As it says at the bottom you can provide a source package if you like, but it's not required. Note that this is specific to closed-source or for-pay apps, anything that is libre and gratis has a different process currently.
As for your other questions:
How is the 32-vs-64 bit thing handled?
You can just upload 32-bit if you like as that works on 64-bit too. If you want to upload 32-bit and 64-bit then build it twice and put the results in "i386" and "amd64" subdirectories.
How is supporting different versions of Ubuntu handled -- 10.04, 10.10, 11.04, 11.10, etc?
You can specify which versions of Ubuntu your uploaded binaries work for as part of the submission process.
How are dependencies handled. If, for instance, my app depends on SDL, how do I specify that?
That will be added as part of the packaging process. If you want to add a comment specifying the dependencies then it will likely speed things up.
If I don't want to distribute source code for my application (which is a proprietary license), is there a clean way to indicate that source shouldn't go with the app?
As stated you don't have to provide the source at all.
If I'm not using autoconf/automake, does my Makefile need to be set up in a particular way, or are the autotools required?
Any build system supported in Ubuntu is allowed. If you are using plain Makefiles then that's fine. However, if you're only uploading binaries that's not an issue.
Best Answer
You've got until Monday to make any changes and submit your app. Any submissions after Monday 00:01 UTC will not qualify for the contest, so you've still got more than 3 days to finish off your app!
Check out the countdown banner on the Ubuntu App Showdown site.