Why does the water bottle not rotate when it is half full? Consider this water bottle: 
When it is full and thrown up in the air, it rotates at a constant velocity. 
When it is less than 1/8th full, the water bottle rotates even faster than when it was full.
When it is half full, however, the water bottle rotates for one half-spin, and then it stops rotating.
Why is this?
By the way, I tested it with a 16.9oz bottle, but the bottles are mathematically similar.
 A: 
When it is half full, however, the water bottle rotates for one half-spin, and then it stops rotating.
  Why is this?

This is why you don't want to ship oil across the ocean in a half-full oil tanker. If you do, you had better equip that tanker with some very good anti-slosh mechanisms. The same goes for trucks, trains, and spacecraft carrying fluid. One quarter to three quarters full is when fluid sloshing (linear fluid dynamics), slamming (non-linear fluid dynamics), and whipping (highly non-linear fluid dynamics) are at their worst. Above three quarters full, most of the fluid cannot participate in the sloshing, slamming, or whipping. Below a quarter full, most of the fluid is free to participate in the sloshing, slamming, or whipping, but there's not much fluid in the container.
In the case of the bottle full of water, there's no room for the fluid to slosh or splash. The angular momentum you impart to the bottle is quickly transferred to the fluid. The bottle and fluid rotate as one.
In the case of the half-full bottle of water, you aren't immediately transferring angular momentum to the water. Instead, your initial flip initiates a slosh wave. That slosh wave is large in amplitude and doesn't have to travel far before it hits the other side of the bottle. This is non-linear dynamics. That wave smashing into the other side of the bottle marks when a good deal of angular momentum is transferred to the water. The bottle's rotation rate slows down markedly at this point (but it does not come to a stop).
In the case of the nearly empty bottle of water, the transfer of angular momentum to the water once again isn't immediate. Once the transfer has been complete, the bottle rotates about a point well below the center of the bottle. Given the reduced mass of the water and the lowered center of rotation, a good share of the angular momentum remains with the plastic bottle rather than being transferred to the water. The bottle rotates faster than is the case with the half empty bottle.
I can't fully replicate your results. I can give a full bottle of water a very hefty rotation rate by imparting some backspin while I toss the bottle. I couldn't make a partially filled bottle rotate anywhere near that fast. The half-full bottle wants to come to a near stop mid-flight unless I crank my arm around a few times before letting go. (This lets the slosh wave hit the other side prior to release.) The near empty bottle does rotate faster than the half-full bottle, but not as fast as the plumb full bottle.
A: I guess, The most important question is wheter you will accelerate the fluid when throwing the bottle. This and conservation of angular momentum will be much more important than dissipation of energy in the fluid.
If it is full, the water, due to the constraints on the system, will be accelerated with the bottle when throwing. So it will just continue to spin.
If it is half full, the fluid will, while you throw the bottle, just flow around not taking angular momentum, once you let go of the bottle the total angular moment will be conserved, but the fluid will be accelerated trying to match the movement of the bottle, as the fluid is much heavier than the bottle, it will seem as if the rotation was stopped.
To confirm this hypothesis, spin up the bottle properly before throwing it when it is half full, then it will continue to spin in the air.
So on a last note, why does it spin faster, when it is filled 1/8? Because then the water will basically just act like a weight at the bottom of the bottle (because at the bottom the centrifugal force will be strong enought to keep the water in place), not leaving the end of the bottle while it is accelerated, thus allowing you to more efficiently transfer angular momentum (compare throwing a straw and a straw with a weight on the end).
A: i dont know if anyone cares yet, but i had this topic more or less at a physicist tournament:
The thing is: once u start rotating the bottle (for the somersault) the center of mass lays a few cm (depending on the bottle) under the waterlevel...therefore the water above (we know it cuz rotations happen aroung the center of mass), it will start sloshing/climbing up the bottle because of centrifugalforces. So then the perpendicular radius increases to the pivot and so does the moment of inertia (I = m*r^2)
so if L (angular momentum stays constant because of newtons law of conservation of momentum) L = I * omega
if L stays constant and I increases, our omega which is the angular velocity, decreases... Thats the same when ice skaters do their rotations and spread out their arms to go slower or pull their arms in to go faster
