Would a white hole have an event horizon? I know, thee are no white holes. My question is not about their existence. I am asking about the mathematics of the white hole solution. Does it produce an event horizon or is it a naked singularity? What would be the physical meaning of such an event horizon? Hypothetically, if a white hole existed, what would it look like?
 A: Here's a Penrose diagram for a white hole:

It's just an upside-down version of the diagram for a black hole. There is a horizon, but the interpretation of the horizon is reversed compared to that of a black hole. The horizon defines an area that no null or timelike curve from past null infinity can enter.

Would a WH spit out random objects like a spaceship with live astronauts?

Yes. 
As in a black hole, the singularity is a spacelike boundary. However, the white hole's singularity is a spacelike boundary that lies in the past of all observers, similar to the big bang singularity. Just as we have no physical laws that can predict what pops out of the big bang singularity, we have none that could predict what would pop out of a white hole singularity. The basic problem is that to make predictions, we need a Cauchy surface, but we don't have a Cauchy surface that lies earlier in time than these past singularities.

is it a naked singularity? 

There are varying definitions of a naked singularity. Most are cooked up in such a way that the big bang singularity is not a naked singularity, and the same would apply to a white hole singularity. For example, one can define a naked singularity as one that can lie in both the future and past light cones of the same observer. By that criterion, this is not a naked singularity.
One of the things that is disturbing about naked singularities is that they break causality by making Cauchy surfaces not exist at all. A big bang or white hole singularity is much less threatening, because it doesn't have this property.
A: Consider the full Penrose diagram for the Schwarzschild metric below. The region I and II are timelike regions similar to the observable universe. The two lines at $45$ degrees are event horizons at $r~=~2m$ The region III is the interior of the black hole and the region IV is the region interior to the white hole. The horizontal lines are the spacelike singularity. Geodesic proceed from the bottom to the top. We may think of particles or fields created by the bottom singularity and absorbed into the top singularity. This is a diagram which represents the solution in a conformal map.
Ordinarily the white hole is ignored. There were speculations back in the 1960s and 70s that black holes resulted in an out gushing of material, but this was largely abandoned. The reasoning being that material falling onto a black hole is accessible in principle by the outside observer. Oscillation modes of matter appear slowed and outgoing photons from matter just above the horizon red shifted by the tortoise coordinate. If this material then erupts elsewhere this seemed to be a violation of mass-energy conservation. 
However, if you think about it Hawking radiation involves ingoing modes and outgoing modes of radiation. A black hole over time loses mass and radiates more rapidly and further in the UV as its temperature $T~\simeq~1/8\pi m~$  increases. A black hole transitions into being more of a white hole. Towards the end the black hole is very hot, bright and explodes. So we might say that in a sense white holes might exist. They come from black holes; we just have to wait around long enough. The Penrose diagram then has some quantum implications.

A: White holes probably don't exist, but they are possible because allowed by solutions of the Einstein equations for General Relativity (GR). There's been lots of speculation and science fiction about them, but the causal aspects are pretty well described by Penrose diagrams, such as those shown by the Crowells postings. 
Whether you define those white hole singularities as naked or not it is clear that the white hole can have energy or information come out to the future. Does not mean a white hole exists, just that it is permitted by GR. It may or may not be possible to have one form.
White holes have been more talked about in the context of what is called an Einstein-Rosen bridge. It is a solution of GR that can be mathematically constructed that has a Black Hole on one side and a White Hole on the other, joined by the bridge. They have never been seen or detected, so it is mostly speculation, but again under some conditions they are permitted by the theory. 
See Penrose diagrams at https://en.m.wikipedia.org/wiki/Penrose_diagram where you can see both Black and White Holes and also wormholes
See wormholes at https://en.m.wikipedia.org/wiki/Wormhole#Schwarzschild_wormholes
The thing about wormhole solutions is that it was shown early on that the bridge between White and Black Holes (or most of them but with the bridge instead of singularities) are not stable, and collapse faster than light can traverse them. That was shown by Wheeler, but it was later shown that if one allows the bridge to include 'exotic' matter (essentially never observed negative energy matter) that the bridge could be stable for long enough that energy and information can go from the Black Hole region to the White Hole region, and escape through its past horizon into a spacetime (sometimes called a universe, but it doesn't mean the two outside regions can not be otherwise connected also). 
So the questions:
1) can a White Hole have an event horizon? Clearly yes, it allows travel one way, from inside the White Hole to outside it. Wormholes allows, possibly, a path from a nearly Black Hhole to another spacetime through an almost White Hole.
2) could it spit out random objects? For White Holes it'll spit out whatever comes out of the past singularity, and there is no way of predicting what it is. For wormholes it'll be some form of what goes in into the Black Hole side. Causality doesn't seem to be violated, there are no singularities.
3) can that cause a-causal behavior? White holes could, but wormholes seems would be causal.
4) doesn't the pure White Hole violate causality in such a way that it would propagate through the future spacetime, and thus not be 'safe', in the same way that naked singularities are hypothesized not to exist? Seems that way. 
The unanswered questions or issues means that classical GR or even quantum fields in a GR background is an incomplete theory, and we need Quantum Gravity to get to what's inside a Black Hole and what is it that exists in those cases that GR describes as singularities.
