Alright, so here's my solution for those who may want it... I omitted several long and boring descriptions of finding your "exact" way through the room that has lots of teleporters and trampolines to make the spoiler shorter and more clear.
1. The first thing to do here is to set up a mechanism that will keep pressing the purple timer. To do that, put one box on the timer, the second one on the stationary conveyor belt. Then push the third box UP through the left blue teleporter.
2. Next, get to the right side of the second room. Try to avoid the cannon fire.
3. Once there, wait until the metal boxes make one turn, then step on the bottom left ice tile. Quickly step up on the yellow button to avoid being killed.
Immediately after that, step LEFT (pushing one box off the mechanism) then step down twice. Wait until the boxes settle down. That stops the conveyor timer from being pressed... temporarily, at least. Go back to the left side of the second room.
4. When the fire passes, enter the left yellow TP from the left, then jump RDD. You'll land in the "last" room -- right in a rainbow coin in a false wall.
5. Push the red spheres down, and wait for both of them to stop. Then push them up. That covers the blue/white buttons at the left side of this room. Now that you have done the part that requires step 3, you need to "undo" step 3.
6. Wait for the cannons to fire and use the trampoline to jump back in the second room. You've previously pushed a box OFF the mechanism to stop it... push the box ON the mechanism (via an ice tile) to start it again. Return to the "last" room via the yellow teleporter's RDD jump sequence.
7. Once there, push the yellow spheres right, and wait for them to stop. Wait for the conveyor at the top to start and stop, then push the spheres up. They land in a water patch. Now stay on the green button.
8. Switch to Loof (hidden in a wall near the green gates), press the cyan button and push the boulder DOWN through the black TP. It will land on the other green button.
Below the now-open cyan gate there's a false wall that contains the second rainbow coin. And because the conveyors keep getting turned off, you're free to go through the cyan gate, and take the two bonus coins as well.
9. Since the green button is now pressed, you can now switch to Stinky and put him on the right red button to open the red gates in Loof's area. Now switch back to Loof.
10. If you look carefully you can see a yellow square button and a false wall below it to provide access. Step on the yellow square button.
11. Once you've done that, go and push both steel boxes in the water patch, right after the yellow spheres. You can now move Loof into the "last" room.
12. Since there was a red timer under the bottom steel box, and Stinky is standing on the ROUND red button, you can now switch to Stinky and step OFF the round red button to open the red gate again.
13. All of the gates are opened, and the conveyor times is perpetually pressed by the moving steel boxes -- now the only thing you have to do is to time your entry across the conveyors.
14. You can now exit with both characters.
(Even if you account for the lengthier descriptions and the formatting (which are purely for clarity) the spoiler is still huge.)
There's a row of false walls behind the push cannons. Presumably they were meant to be used, but this solution does not require them.
The complicated mess of teleporters and trampolines in the second room is presumably to be able to jump behind the cannons. For those who're interested in how to do that, here's a sequence that (after doing step 3 to turn off the mechanism that keeps turning off the conveyor... or just timing it exactly) allows you to get behind the push cannons:
Wait for the cannons to fire and enter the left yellow TP from the left. From there, jump RDURRDLU to get behind the cannons.
If you jump RDURRDLL instead, you can get to the bonus coins even when the conveyor timer isn't being pressed by the steel boxes.
Please note that this is MY way of completing this puzzle. I doubt John925 intended us to finish the level without push cannons, and I highly doubt I've used the "right" buttons. But this way I've described does allow you to get the gold star, and that's what matters, right? Especially seeing as John925 has not been seen for quite some time...
).
Blessed are those who can give without remembering and take without forgetting.