Schrödinger's cat is a thought experiment which strains the capabilities of the interpretations of quantum mechanics as they have to deal with macroscopic objects that are described as being in a superposition of states. If you run the math on this experiment, you get.... the math on this experiment. You get the solution to an equation in the form of a wavefunction.
It is up to the interpretations of QM to assign meaning to those equations, and it is these interpretations which get strained.
The Copenhagen interpretation declares that when you open the box, you "collapse" the waveform to a concrete state where the cat is either alive or dead. It states that whether the the cat lives or dies is randomly selected.
Now if we really poke hard at what these interpretations say, you really have to use a slightly weaker wording. We say that the state of the cat when we open the box is indistinguishable from a random result. That phrasing opens doors. It is entirely possible that there is actually some supreme being which has a book which covers the entirety of time, past and future, and in that book it states that the cat lives. However, without this omnipotent knowledge, this is the best we can do.
The Many Worlds Interpretation is the one which discusses the idea of the universe splitting. Unfortunately, that was not a very convenient word choice. It got popular, because it had a certain visual appeal. The interpretation more specifically states that the observed values (such as the liveliness of the cat) are dependent on the subject, not just the object. In MWI, observations are not objective truths. The mathematics for this superposition permit treating the universe as if it were two parts, one with an observer that sees a living cat, and one that sees a dead one.
Pilot wave theory does not have randomness nor universe splitting. It has a universe-wide wave function and particles that interact with it. In pilot wave theory, whether the cat is dead or alive is deterministic, but it depends on the state of every object in the universe (even those outside of the light cone of the experiment). As such, it is unknowable whether the cat is alive or dead, until you open the box, even though it has a definite value.
Each of these interpretations has to "give up" something that we hold dear in the classical world in order to reconcile the real experiments done in the field of QM. Copenhagen gives up determinism, letting naked random variables go scurrying around. MWI gives up objective realism. Beauty is truly in the eye of the beholder in MWI, because the state of the observed can only be stated with respect to a given subject. Pilot wave gives up locality completely, letting every single "decision" made at the quantum level depend on the state of the entire universe.
As for whether there exists a truly random thing, consider that a random variable requires there to be meaningful mapping into a probability space. In a lot of cases, with exactly 1 world, where events happen exactly 1 time, this definition is fraught with difficulty. A less strict reading would be to talk of "nondeterministic" behavior rather than random behavior. All random behavior is nondeterministic, but one can consider non deterministic things (such as minds) which are not random. The philosophical discussion of determinism vs non-determinism is well documented, both in other questions and in the literature at large.