Electrons jumping from 1 water molecule to another
Electrons can be anywhere in the universe 


Kind of: There is a non-zero probability that an electron can be at any distance from some point of origin (for instance, the nucleus of an atom), but there is some distance which corresponds to the place where we're most likely to find it (the place where the probability amplitude is highest) most of the time.


  and where they are is random.


Again, sort of: They are most likely located in certain regions of space for most of the time, but since it's a probability, it's not a certainty. That said, there is a common sense element that can be applied here. Yes, an electron from a hydrogen atom in a flask in your laboratory can be out near Jupiter, but realistically, it's not. And if it's on Jupiter, it's not really that hydrogen atom's electron anymore.


  Is it possible for an electron to jump from 1 water molecule to another causing $\ce{H2O}^+$ and $\ce{H2O}^−$ to form?


Yes, it is possible (but probable? I don't know, but certainly there are conditions under which such a reaction will occur).