I think I finally managed to figure out the thing (after taking it entirely apart only to realize there was nothing wrong with it in the first place (I think)

The timer button is a 1 minute delay. It does make a few beeps but the sound is hard to hear since the charging sound of the capacitors is so loud. It is a simple timer for processing time like you said. 1 very weak beep at 30 seconds, 1 longer one at 1 minute. Need more time, press the button for a second sequence.
Now for the light... it's hard to explain... It is a flash, (or else I don't know what the very large capacitor inside would be for), but it is not instant light like a flash. The burst lasts for about ½ a second. There is a 6 step exposure adjustment (3 lighter, 1 OK, 2 darker)
Now, for what really got me stumped... there is a switch in the slide slot. If you don't put a slide in, you can't accidentally expose the film as it locks out the print switch. The printer came with a type of slide holder which is cut so that it won't trip the switch. And if you put a slide in it, you have to make sure it's thick enough (which my test frame isn't).
When you look at it closely, you can really see that the guys who built the thing went through a lot of trouble to make it complicated. The circuit board is huge and loaded with stuff. The optical path is very unconventional (there is a light sensor on the side of the box with what seems like a mirror assembly... (I hope my spray contact cleaner job didn't wreck it)...
Anyways, sweet piece of Polaroid work. Now all I need is a few slides and a pack of film...