I have no clear idea what a probability of event happening in the Omniverse means.

You obviously aren't familiar with the concept (for which I cannot be held accountable). In any event, I'll explain it briefly: the omniverse is that state of affairs in which all possibilities are realized. Hence, that any event should obtain therein is an absolute certainty.

The word 'omniverse' does not represent a recognized concept in mainstream physics. You give a definition for it and implicitly imply in your root comment that MWI implies the existence of an omniverse,

Consider that on the scale of the Omniverse [...],

but provide no justification for this.

What is "long run"? Does it mean "in other universes" (that would make sense, but the choice of words "long run" to denote that seems bizarre) or does it mean "sometimes later in this universe" (that would be the natural interpretation of "long run", but then the statement says "p(the event happens) = 0 and the event can happen", which is a contradiction).

This is the standard understanding of what objective probability teaches: given any universe you please, a given probability of an event is supposed to hold for a particular situation in the case that one were to observe all cases (for all time). Thus, the "long run" considers a particular situation for all time. If you flip a coin, you will not observe an outcome of 50% heads and 50% tails, but were you to flip this coin for eternity, the net result is just such an outcome.

You are either using different definitions of "long run" or making contradictory statements across your comments. Here you state that this specific outcome will obtain in the long run, but in your root comment you argued that any alternative outcome can obtain in the long run:

[E]ven if the probability for an event in our universe were 0 that would in no way serve as an impediment to its occurring in the long run.

And of all that, how does anything imply, or even relate to, the "uncountably many" answer you gave at the beginning?

I'm not sure how you'd pose this question seriously. For one, the MWI and nature of QM decoherence shows a state of information as unrelated instances of a general state of things (in a coherent superposition). That there are uncountably many universes (as inhabited by any observer you please) in which the cat is alive, and so too for the cat being dead. The "cat" could even be an infinite variety of other objects, for all we damn well know.

You use the phrase "uncountably many" without providing justification. I can substitute this phrase with "countably infinitely many" and the structure and strength of your arguments would not be changed. Provide justification for why the cardinality of the natural numbers is not enough. (This was Prase's point in eir reply to your root comment, which you did not address.)

I even don't understand what do you mean by "taking the universe as a QM event".

Then you obviously aren't familiar with MWI. Even Penrose and Hawking agree that QM applied to the universe implies MWI.

This response is not constructive. Provide references. Also, you changed context from

taking the universe as a QM event

to

QM applied to the universe,

without clarifying what you meant by the first phrase, which I cannot parse in a way that makes sense.

From the single sentence the OP consists of, could you quote the section where it very clearly asks for non-standard instances of the (which?) question?

Are you being obtuse to justify your down-vote or something? This is ridiculous. Now I have to justify my answer to the OP to you? Absurd. But I'll play along, quoting OP:

...what about non-50/50 scenarios...

I think that the universe is a "non-50/50 scenario", but I guess you can make the case it isn't.

First, your tone unnecessarily escalates the hostility in this comment thread. Second, a binary quantum event in which the final quantum states have unequal probabilities is standard. It is the opposite, where the probabilities are equal, which requires specific preparations, because we would need to make sure that the square of the modulus of the probability amplitude of both states are equal.