Let's start with Schrodinger's cat. The whole point of the hypothetical is to make a point. The situation: A cat is in a box with a capsule of poison gas that may break open at any random time. There is no way to see into the box or determine the state of the gas or the cat. Therefore, until we open the box and examine the cat, we cannot know whether or not it is alive or dead; there are two possible situations, but only one can exist. The cat can be said to be both alive and dead; it is only by looking into the box that we can know whether it is or not.
It has nothing to do with a god tossing dice or whatever. It's simply a fancy way to make the point that there are situations in which something has no defined values until you measure it. Think about a double-slit light experiment. Depending on what values you measure, the photon that you fire at the wall with the double slit will either go through both slits (experimentally confirmed by interference patterns) or one slit (experimentally confirmed as well by lack of any interference patterns). Now then.
Shrödinger's Cat tought experiment's point was to make a point that it is silly cat can be considered both dead and alive at same time. While on quantum level, probabilities make sense, when applied on newtonian scale, it's extremely counterintuitive, and the whole thought experiment was made to point out exaclty that.
And yeah, it does have to do with God/universe/etc tossing dice. When you measure a wave funtion, what determines the state it will "collapse to"? It follows probability, but if the probability is caused by something, the process is deterministic. If the probability is just flat out non-deterministic mathematical probability, what decides the outcome? It implies God throwing dice, as we pull out number from nothing, and apply it to real world.
As for what I assume you're referencing with regards to stuff not needing to be in contact with other stuff, the Einstein-Podolsky-Rosen paradox, their conclusion was based on an incomplete understanding of causality from several decades ago, and furthermore, classical physics concepts do not necessarily correspond to reality. Perhaps quantum mechanics does not either, but in all situations, QM approximates reality better than classical physics. In addition, Bell's theorem is not the only way to solve that paradox.
The authors' preferred explanation for EPR-paradox was that the result of wave function collapsing was "encoded" in the particles. Which would be totally in line with relativity. But Bell's Theorem didn't solve this paradox, it simply made the preferred explanation impossible. Bell's theorem came to conclusion that no physical theory of local hidden variables can be used to explain the predictions made by Quantum Mechanics. Either the locality or the reality of the explanation would have to be false. Non-locality would allow for FTL transmission, and so violate special relativity(as going faster than light would violate causality according to it).
Non-reality would mean that objects do not exist or have definite states independently of the observer. This would be the idea Shrödinger found ridiculous, and then he deviced this whole cat killing thing to point it out. Wigner's Friend thought experiment is similar and shows how two people can perceive the same wave function differently. Also the whole thing lead to another famous Einstein quote, “I like to think that the moon is there even if I am not looking at it”.
So yes, it would certainly seem that classical physics concepts do not in fact correspond to reality. Especially considering the whole "reality being screwed up if we actually believe in predictions of QM" was resolved in favour of QM.
Einstein was brilliant, but brilliant minds aren't always right. Einstein himself said that in general relativity, his greatest mistake was creating a cosmological constant, and Einstein was also rather convinced in a static universe; a universe that does not change, which we now know to be untrue.
First time I brought up Einstein was due to Mage very intrestingly thinking me as a man who thought that for sure hidden variable had to exist, and then smelled Nobel in my future. The second time, as I don't believe anything can be 100% pure mathematical probability, unless it really is the God itself throwing dice, generating pure, non-deterministic numbers out of thin nothingness and applying them to our reality. And then it was as a reply to rebelsquirrell to further emphasise how Einstein would've much liked local realist universe, which is in odds with Bell's Theorem.
ANYWAYS - The point was, there are several different interpretations of Quantum Mechanics. Some of the interpretations, such as many worlds intepretation(which is one of the most popular QM interpretation with actual theoretical physicists) and pilot wave theories are deterministic.
Of the other interpretations, "shut up and just calculate" is the one that is probably the easiest to apply to pratical problems.