What will happen with the liquid inside a falling bottle when the bottle is suddenly stopped? When I dropped a bottle with a small amount of olive oil left in it, the oil got launched out of it when it hit the floor.
Now imagine we perform an experiment. We fill a bottle with a liquid. We let it hang on a piece of string for a whìle so the liquid can relax. When the piece of string is cut (or burned to prevent motion in the liquid), the bottle will fall on the surface beneath it. Say we let it fall in transparent cylinder in which it just fits.
What will we observe? Can we say anything in advance? Will a central ray of liquid shoot upwards? What influence will all the variables have? For example, will viscosity, the dimensions of the bottle, the height of the liquid, or the height from which we let the bottle fall have a significant influence?
It seems that every time we drop the bottle the result will be somewhat different due to the chaotic quality of the process. So we'll have to take some kind of averages of results coming from quasi-identical initial configurations.
 A: Liquid slosh and splash can be weird, non-linear, and chaotic, particularly so in response to a significant impulse, which is what you are describing. Entire books have been written on slosh and splash. Slosh can be a big problem in space vehicles; slosh has for example been cited as a cause of the failure of SpaceX's second launch of its Falcon F1. Slosh has caused the failure of multiple spacecraft, and also has caused ships at sea to capsized, has caused airplanes to lose control, and has caused trains and trucks to overturned.
You have an open container and are subjecting it to a significant impulse. This makes the physics even more complicated. With regard to variables in your experiment, the viscosity of the liquid, the speed of sound in the liquid, the density of the liquid, the level to which the container is filled, the flexibility of the container, the shape of the container, the shape of the opening, the height of the drop, the softness (or lack thereof) of the landing: all of these (and more; I'm sure I'm omitting some factors) will come into play.
I suspect that increased viscosity will reduce the response. I suspect half full will sometimes result in the biggest splash because with that both slosh and splash come into play. It is chaotic, so sometimes you will see a reduced effect with a half-full bottle while other times you'll see a large jet.
