Is this statement always true (= a tautology)? Nothingness here does not mean absolutely nothing, instead it means: nothing but logical laws. I'd formalize it as ∅ |= ◻ ◇ F (F stands for any wff) or maybe ∅ |= ◇ F and as far as my modal logic knowledge goes it is a valid semantic consequence. Am I right?
Following from your comments, I think your question is not so much about whether it is logically necessary that something exists, but about whether probability theory can be used to demonstrate that it is highly probable.
To start with the question as you have asked it, logic itself cannot tell you whether something exists or not. In the standard way that first-order predicate logic is set up, we assume that something exists. This is implicit in the way the rule of universal instantiation works, and corresponds within model theory to the assumption that the domain is always non-empty. If we wish to reason about potentially empty domains, we have to use a non-standard logic, viz, one of the free logics. Even then, the logic itself does not tell us what exists and what doesn't.
Using modal logic does not change anything. A thing might possibly exist, in the sense that it exists in some possible world. But, the logic itself does not assure us that if a thing is possible then it is actual. If we wanted to be able to reason in that way, we would have to make use of some assumptions to that effect and incorporate them into the logic. Either way, they are still assumptions.
You say in a comment, "...probability theory ... does not belong to logic". Whether probability theory should be regarded as part of logic is not the issue here. The important point is that Carrier is abusing probability theory itself. He says that since there could be any number of universes from zero to infinity, and each of these possibilities is equally probable, the probability of there being no universe is (approximately) zero. This is nonsense. One cannot have a proper uniform probability distribution over an unbounded range. Sometimes, in Bayesian theory, one can proceed from an improper prior to a proper posterior, but that is only feasible when you can update on known conditional probabilities.
In the case of the existence of the universe (or universes), there is no way to get a handle on such things. We don't know anything about the probability distribution of possible universes, or whether we, or anyone, would be around to observe them if they existed. It is precisely this that makes it impossible to assess things like the probability of the universe existing, or the probability of God existing, or whether the fine tuning argument tells us anything about either.
As a corollary, note that Carrier's argument, if it were sound, would also prove the existence of gods. Since there might be no gods, or one god, or two, or three, or any number up to infinity, then if we assume that all of these possibilities are equally likely, by Carrier's reasoning, it would be overwhelmingly probable that some gods exist. Not a conclusion that he would be happy to endorse!