Fermenting Sauerkraut - Should I Stir?
Well, having grown up near "the" Sauerkraut region in Germany - I'd say don't. 

Honstly, I hadn't ever thought about why until today (can't have been only lazyness that my ancestors left the kraut in peace until done.), but:

Why making really sure to create a water-seal when you are breaking it with stirring? The kraut is supposed to ferment under the absence from air! You are supposed to check on the kraut regularly and even fill up with saltwater, if if the water level is too low. If even the smallest bit of cabbage sticks out from the brine, you risk mold. 

Besides, I'd be afraid to drag something nasty into the half-done kraut (and I'm so not implying that your utensils aren't clean!).

I'm adding a detail from a classic fermenting pot: note the groove at the top which is supposed to be filled with water. The lid sits in this, ensuring the seal. The two stones are supposed to weigh the kraut down (what you do with the second plastic box).