On the water-splash, here's a video that I think explains what happened. https://www.youtube.com/watch?v=2UHS883_P60 (the 3 balls, not the double bounce on the trampoline).
The Newton law of conservation of energy says that energy is concerved, so a share of the energy of the drop transfers to the energy of the splash.
Figure you drop it from 1/2 meter and the mass is 0.2 KG (about 7 OZ). Do a little math and 1/2 * 9.8 * .2 = .98 Joules, or about 1 joule of energy.
How high a splash can 1 joule make in still water, well, it depends. The energy is converted into 3 different types of energy, wave energy, which is likely the biggest share, splash energy (#2), which depends on a few factors like the shape of the object, depth of the water, if there are already waves present, etc, and the 3rd and likely lowest of the 3, heat energy. Here's where the YouTube video comes in. If it splashes just right, you can get much more velocity out than you put in, into a much smaller mass.
There's no good calculation that I know of to give you maximum height a water drop can reach in this scenario. It depends on how well the energy transfers into a drop and what size drop.
As you know from experience, this kind of splash-back is rare, but dropping from a lower height or dropping one at a time vs 2 at the same time should reduce the likelihood of a high splash - that's probobly super obvious.
A curious and loosely related field of study is wave size from large meteor impacts or landslides into the ocean. A rather fun one to read about is the Canary Islands waring, that one research team made and others think they exaggerated, but it found it's way onto television and there were warnings that a Canary Islands landslide could create 50 foot waves across the entire eastern seaboard". Probably not accurate, I think, but the warning made some impressions all the same. More on that here: http://news.bbc.co.uk/2/hi/science/nature/3963563.stm
Firstly, there is a relevant physics stackexchange answer here: Why is the splash of water always vertical, no matter in which angle the gunfire hits the surface? That was for a gun firing onto the water surface.
You mentioned two cases: Placing a glass of water and spitting into a body of water.
In the case of placing a glass of water on a desk, decelerating the glass rapidly before placing it on the desk, you cause disturbances on the water surface which result in waves. Sooner or later, the right combination of waves cause the water to break free from the surface and cause a splash as seen in this video.
As for spitting on the water surface, the initial impact of the spit droplets causes a reaction that pushes water back upwards after the droplet has landed. The temporary void or hole cause by the landing of the droplet results in a circular wavefront rushing inwards to fill the void. As the wavefront meets, water is pushed upwards, sometimes causing a spectacular splash, as seen in this video.
Best Answer
The other answers may be correct but claims that the solution to this is "obvious" for a real fluid are, in my opinion, questionable.
However, people have studied similar problems empirically. See for example the 2009 paper "Experimental Splash Studies of Monodisperse Sprays Impacting Variously Shaped Surfaces" by Yoon, et al. (DOI:10.1080/07373930802606188).