- Where am I wrong in my assumption about natural water distillation (i.e. is rain water not as clean as it should be in theory)?
Generally, rain water is pretty clean. That is not an issue. Plus, we treat ground water and reservoir/lake/river water, so we could easily treat rain water. What you really fail to miss is that ground water and reservoir/lake/river IS just rain water. It is the water from a LOT of rain. When it comes to water use, humanity is taking water faster out of the ground than the ground can collect it. Now, if the ground can't collect water fast enough, and the ground covers nearly all of the land (except buildings and bodies of water), then obviously we humans can't practically collect anymore rain water than the ground. So, as long we use up more ground water than the ground can "collect" due to rain fall, we're going to run out of water. Now, there are some technical ways we could collect water better than the ground (e.g., don't let it evaporate, don't let it runoff into unused rivers). However, there is no way we could get as much water as all the land on Earth collects!
- Why don't we collect rain water (at least for plant watering, if it is not suitable for drinking)?
We do collect rainwater. We normally collect rainwater when it is easier to collect it locally than to get it from where it naturally flows to. If you count dams, we hold back water and collect water that way so it doesn't flow too quickly to the ocean, where it becomes too salty to drink or use for irrigation. Some people also collect rainwater from their roof, while others just leave out an open barrel (that's not much of surface area for collecting rain).
- Theoretically: after an animal consumed some amount of water, say a glass, how long will it take, given that animal just urinated on the ground, for that water to go through water cycle and become ready for drinking again?
That depends on if the urine is going eventually into a river, shallow groundwater only 50 feet (or less), or a subterranean aquifer a few thousand feet down. It can evaporate immediately from the ground or flow into a river and evaporate the same day. On the other hand, some groundwater percolates far down into the Earth into "aquifers". We remove water from those using wells, and we say those aquifers "recharge" when more water moves into them.
The famous Oglalla aquifer recharges at a rate of about 0.85 inches per year. That is, if we draw it down an inch in a year, it won't even fill up to where it was in the next year. How long it takes water to get to a specific level from the surface is affected by porosity and the amount of water already in the rock. Water can't flow when it is blocked by water, right? Also, much water also moves horizontally as well as vertically. So, asking how long it takes for water to go from X to Y directly below it would assume that water even follows that path. It might be that Y only gets water from a far-away location Z. So, I will pass on the "how long" part of the question. I will add, though, that some groundwater is tens of thousands of years old. There are 12,000 inches per thousand feet. If an aquifer is 3,000 feet below the surface, it is 36,000 inches down. If the aquifer recharges at 1 in/yr, then dividing the distance (36,000) by the speed (the average flow of 1 in/yr) give you the time of travel -> 36,000 years. Again, that is an idealized vertical drainage with no feedback mechanisms and constant flow rate. The situation is more complex than that, but that's how you can think about it. :-)