It should be possible. Sites such as Kongregate are permanently connected to the internet through Flash. Perhaps you need to run the Flash application inside another Flash application, where one will send stuff to the internet.

And with themes, all the objects will look slightly different in an ice cavern level than inside a city level.

Are you talking about the stuff they save through cookies? I don't think that really counts...

Not cookies, stuff such as achievements and the chat. They have this Flash window with the chat, leaderboards for the game and the possible achievements you can get, and in the middle of that window the actual game plays. If you get the requirements for an achievement in that game, it automatically updates your online profile. They also have something that makes it possible to share levels and stuff without text.

Where in that mess does it store and retrieve files, locally?

If it's not possible to both connect online and locally, it looks like Aeon will be online, while the level editor will be offline (containing only the engine from Aeon, with on-the-fly checkpoints?, for playtesting), and maybe a link that opens the Interguild search if you click a "Find more objects online" button, or something, and then you have to copy paste to and from.

You can see it yourself here:
http://www.kongregate.com/games/Pixelante/pixel-legions

I don't know how it works, but it appears to connect both to your computer and to the internet.

Lol.
I'm sure whatever it is, it's gonna take forever to load.

There's no reason why we couldn't allow them to use any themed object within any theme. Doing so would be unnecessarily limiting. If you want a new theme, you go to the tile list and hit pgdn, or something.