I'm pretty sure the answer is no, but I figured I'd ask.

What are you asking? Your not giving much to go on...

You can play any sound you want from a script. But its easier to create a new sound object and set it to loop. Then drop the sound object into the world where it will play continuously.

I have 30 songs that play on my jukebox. I have 30 sound objects, all set to initially disabled. When I want a song to play, I enable it and that song plays. I can also disable it at any time and it will stop playing, and I can enable a different song.

No, for objects that have a looping sound property, such as http://geck.gamesas.com/index.php/Activator or http://geck.gamesas.com/index.php/MovableStatic, I'd like to have control over that property from script, such as being able to get or set that http://geck.gamesas.com/index.php/Sound reference. I'm fairly sure I'd have to wait for new FOSE functions designed to do that though.

But more specifically, all I'm really interested in being able to do at the moment is to stop and start the looping sound on objects that already have them. Has anyone maybe found any tricks that lets you do that?

Still not very clear. Do you mean the sounds that are attached and assigned 'inside' the NIF itself?
There is nothing I know of that allows you to control that. Its usually controlled by the NIF which is probably hardcoded in the game engine.

No... Maybe I need pictures. Many objects in the game have looping sounds assigned on their forms in the GECK, such as Activators (http://stashbox.org/838028/ACTI_form.png), and Movable Statics (http://stashbox.org/838031/MSTT_form.png). I'd like to be able to stop or silence those on individual objects while in-game, and also be able to make them resume.

Make a copy of the object that has the sound, but on your copy, delete the sound.

Then place the silent object in the exact same place as the original. But set the silent one to initially disabled.

Then, in a script you can enable the sounded one and disable the soundless one, and vice versa to make it silent.

Right - if anything, that's the kind of thing I'll have to do. I was hoping to avoid that, since I'm not talking about a single object here, which was the reason for this thread.