Very good questions. You're obviously thinking about this quite a bit.
There are a couple levels to the discussion of randomness. You're absolutely right tht if we know everything about a coin toss with sufficient precision, we can predict the outcome. In fact, coin tossing robots have been built, often for use in statistics simulations by students, which are so accurate that the coin always lands the same way. So, the randomness of a coin toss is really a reflection of the slight variations in human movement. So, it's not really a random event, it's a poorly calibrated one which gives rise to a sort of pseudo-randomness, if done sloppily.
Then there's the question of randomness at all, and does it even exist in nature, or is nature, as many early scientists thought, entirely deterministic. This is a much more interesting question. At the large scale, that of macroscopic objects, it appears to be deterministic, or at least sufficiently close to deterministic for most cases, that the early laws of physics were all similarly deterministic. In the early 1700s, many though that if you knew the position and the momentum of everything in the universe, you could predict the future accurately. This "clockwork universe" idea was very much a product of its time, and as we learned more, it became apparent that there was true randomness.
The quantum scale is where we find truly random events. It's possible to make probabalistic statements about quantum mechanics, but it's not possible to say exactly where a photon will land, or precisely where an electron is at a given time. This was debated at first, and led to the development of a branch of quantum mechanics (QM) that tried to show that there was determinism, but we just didn't (or couldn't) measure everything with sufficient precision. This was called "hidden variable theory". In the 60s, this issue was finally put to rest when the theory was developed to the point where there was found to be a measurable difference in the hidden varaible theory, and other pursuits in QM which accept things as random, even if the additional varaibles are not measured. This was JS Bell's inequality, and what it is and what it means is too much to go into here. Look it up if you want to know more.
So, it looks like at the quantum scale there really is true randomness. But, this can be difficult to interpret. The idea of universes splitting is actually one interpretation of QM, called the Everett-Wheeler hypothesis. The more standard Copenhagen interpretation involves a single universe, but a non-strongly deterministic model of QM. The EW hypothesis, also called the "many worlds" interpretation is really very interesting and has a lot of good arguments going for it, in my opinion.
Neither hypothesis has any provable outcome that can distinguish them from each other, but a recent poll by the American Insititue of Physics found that among theoretical physicist, EW hypothesis and the Copenhagen interpretation have roughly equal popularity as far as which one "seems right". Perhaps as science progresses we'll be able to understand these ideas well enough that we can experimentally determine which is right, but for now they're both open questions.