What happens to the air in a water bottle when it falls? (ignoring the air resistance that causes the bottle to take positions in mid air. Let's say it is dropped in a vacuum with earth's gravity accelerating it downwards) I just cannot catch what happens with the bubble with my bare eye, and I thought I should better ask some experts :)
 A: If you drop a bottle without any residual motion (that is, it is not spinning etc) then everything inside that bottle will be in "free fall". The air and the water will attempt to fall at the same rate.
There is a nice video of what happens to water when it "spills" in the International Space Station: it becomes a "blob" because of the surface tension. Add to that a small amount of attraction between the liquid and the bottle it is in (hydrophilic surface - attracting rather than repelling water) and you see that the water will actively want to stay in the bottom of the bottle: there is no force to move it away, and there is actually a (weak) force keeping it there.
So the air stays on top, and the water at the bottom. When it hits the ground, it will make a big mess - but that's not what you were asking about...
A: I think nothing. Example #1: Calculate the rms speed of an oxygen gas molecule, O2, at 31.0 °C
Solution:
$v=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3 \times 8.31447 \times 304.0}{0.0319988}}=486.8 m/s$
That is pretty high average speed of oxygen molecules. The acceleration of the bottle does not change it much at beginning. The direction of molecules motion is random if the speed of the bottle is much less than computed above number, but with time they will slow drift up relative to the bottle. 
A: The environment in a free-falling object is the same as one without a gravitational field (neglecting gradients in the gravitational field, i.e. tidal forces).
So, before you drop the bottle, gravity is holding the water at the bottom and creating a slight air pressure gradient between the top and the surface of the water, and a water pressure gradient between the surface and the bottom.
When you drop the bottle, gravity "vanishes." The pressure gradients disappear. The air at the surface pulls upward a bit as its pressure falls. More significantly, if it's not a rigid material, the bottom of the bottle will rebound a bit because it's no longer counteracting any water pressure.
The net effect is that, even without air resistance or prior water currents, the water will tend to start sloshing and mixing with the air.
A: A liquid / gas mixture in an enclosed vessel in a zero-g environment can be quite weird. Given enough time, the contents of the tank eventually become an ethereal mix of blobs of liquid, bubbles of gas, and foam. This creates a significant design challenge for liquid fuel tanks intended for use in space. One solution is to use a bladder to separate the liquid from the ullage gas. That's no fun! There is no mix of liquid and gas in such a tank. (Plus the bladder is a point of failure.)
Sans a bladder, the only ways to get fuel from the tank to the thruster are to never stop firing the thrusters (thereby forcing the fuel to one end of the tank), rotating the tank so as to centrifuge the fuel, or to use devices (typically passive devices) based on surface tension that collect the fuel. These are called propellant management devices, and there are lots of patents for such devices.
A: In the short term, the sloshing of the water as you not-completely-cleanly let go of the bottle will be the dominant effect. In the long term, since you're ignoring air resistance, and if we assume the bottle isn't spinning in any way, then surface tension will dominate.
Different material interfaces have different energies, and absent other forces those energies will try to minimize. For instance, all else being the same the water and air will rearrange to make the air/water interface as small as possible. Depending on the plastic (and its wettability) the plastic/air or plastic/water interface may also minimize to varying amounts. (I took 3.091, "Introduction to Solid State Chemistry", from Professor August Witt at MIT in 1980; what an awesome lecturer!)
For example, if the plastic is extremely hydrophobic, then the water/plastic interface will be minimized and you'll end up with one big ball of water floating in the middle of the bottle. Or, if the plastic is extremely hydrophilic, then you'll end up with a big ball of air floating in the middle of the bottle. 
