If I hang one 5 gal. Bucket directly above another, put a hole in the bottom of the upper bucket with a tube inserted just inside hanging all the way to the bottom of the lower bucket and fill the upper, will it drain completely into the bottom bucket?
Yes, at least if you ignore the little droplets that leave the bottom of the upper bucket a little damp. There will also be some water in the tube, but it won't be much higher than the level of the water in the lower bucket. (There might be some capillary action that raises it a little, but probably not much.)
I suspect the reason you ask this is because you wonder if the pressure at the top of the tube will be lower than the pressure at the bottom of the tube -- because the bottom will be immersed under a bucketful of water, but the top of the tube will have no pressure. And indeed, that pressure difference will be there. So it might seem as if the water should want to drain from high pressure to low, and thus up the tube.
The reason it doesn't do that is because of the weight of the water in the tube itself. Once the draining is done, the pressure just inside the bottom of the tube will equal the pressure just outside it in the lower bucket, which means that the water won't move any more. The source of the pressure in the tube is exactly the same as the source of pressure in the bucket: the weight of the water above.
Yes, provided you wait an infinite length of time :)
First, forget the tube. It has no effect on the problem, except maybe to minimize splashing.
Assuming the rate of flow out of the top bucket is proportional to the pressure at the hole, you can write a 1-term linear differential equation for the amount of fluid in that bucket, for which the solution is an exponential decay.
In the field of Pharmacokinetics, the flow of drug compound from gut to plasma to/from tissue to/from liver and kidney are modeled in terms of "compartments", where compartments are essentially the same as your buckets, connected by tubes.