All things happen for a reason

No fallacy here, but certainly an interesting philosophical/psychological question posed. First, it is important to define "reason" in this context. Are we talking about a psychological state that precedes an action or behavior, or a cause in a deterministic chain of events? If one believes in the concept of "free will," is an act of free will considered a "reason?" This is all in preface to the question "do all things happen for a reason." The question "Could someone murder someone with zero reason?" is easier to answer. No, they can't, since the definition of murder requires premeditation, or a psychological "reason," even if the reason is "I am a psychopath and I like killing people."

"Could someone murder someone with zero reason?"

I believe the answer is yes, you can murder someone with no reason. Because "reason" is short for "reasoning". 

You could just act based off a triggered emotion in the moment, no reasoning happened in that case, but reasoning could happen after the murder to try and justify it, or condemn it, and or try to figure out why the murder happened. 

So the question of whether someone could murder without any reason hinges on what reason means. You need to be careful to avoid using different definitions interchangeably as that can lead to equivocation.  

Reason can be a verb meaning a thought process, or it could be a noun meaning a justification or motive. These two are linked as one would provide the other, however it could also be a noun meaning an explanation or cause. Often the context will help narrow it down e.g. if you had said "without a reason" that would rule out the verb. 

If you're just going with explanation or cause, then yes, there must be a reason. You may not know what the reason is, but that wouldn't mean that there isn't one, that would be a form of argument from ignorance, where a conclusion is erroneously based on a lack of apparent explanation, rather than accepting that it's unexplained.

And if you mean that the person has a justification or motive, murder requires premeditation which would mean that the person thought it through in some way. 

Bo is more qualified to talk on psychology so might have something I didn't think of or know about. 

Just to add, I know someone who says everything happens for a reason and I'm pretty certain, though now I'm thinking I should actually check with him, that he means that there is a purpose i.e. a thinking agent behind everything. If he just means a cause, then yeah, obviously....

There are no fallacies in what you've said, but there are potential fallacies as highlighted.