If the universe is infinite, would QM allow the existence of "weird" zones? If we suppose that the universe is spatially infinite and extends more or less homogenously in all directions without end (having similar galaxies, stars etc.), then we can assume that there is a more than zero probability for a macroscopic quantum tunneling event to occur somewhere (let's say a pebble tunneling through a barrier/rock wall or something similar). But if the universe is infinite then there could be infinite such tunneling events going on at any given time in different locations. In such a case if we consider the set of all those events where such tunneling occurred, we can ask further whether if another tunneling event occurred just after the first one. Surely the probability for that (two macroscopic tunneling events occurring sequentially or side by side)  would be even smaller but that still would be more than zero (think of the pebble tunneling back to the original side just after tunneling to the other side). Hence such set of two nearby or sequential tunneling events would also be occurring somewhere in this infinite universe.
Taking this logic further and increasing the number of simultaneous or sequential nearby  tunneling events we can think of an entire "zone" of the universe where this tunneling occurs all the time macroscopically (just because it can due to still having a miniscule but non zero probability). Does this seem to imply that there would be zones in an  infinite universe where apparently the normal natural laws seem to be violated just from the implications of quantum mechanics (which in itself is a set of normal natural laws).
What would observers in such "weird" zones conclude about the nature of universe and its laws and would they be able to develop similar theories of QM (an experimenter inside this zone might conclude that macroscopic tunneling is actually the 'norm' in nature and might derive very different set of laws from that). What about the possibility that we ourselves might be living in such a weird zone where laws which we consider to be the norm from our perspective are actually quite rare and exceptional phenomenon when viewed from the perspective of the universe as a whole?
Does this scenario seem plausible and is the existence of such zones very likely in the case of an infinite universe governed by QM?
 A: Statistics can be tricky.  This is especially so when you involve infinities.    So the first thing we will need to do is break down what it means for an observer to develop a scientific theory in their "weird" area.
Probably the best tool in your pocket is Bayesian Inference.  You are probably more familiar with frequentist inferrence, which is typically what we are taught in school.  Both Bayesian and frequentist approaches to statistics use the same fundamental math of probability, they interpret the results differently.  Frequentists infer the frequency of events occurring.  Bayesian inference can be more thought of as dealing with the "belief" that something is true.  Consider a 6 sided die.  You roll it a few times, and observe that it comes up {6, 6, 6, 6, 6, 6, 6, 6}.  Do you believe this is a fair die?  That is Bayesian thinking.
In Bayesian inferrence, we talk about priors.  These are your pre-existing beliefs.  Then, with every observation, we update them using Bayes rule to get posterior beliefs.  You may initially think the die is fair, but after observing a few 6's, you would start questioning that belief.  Bayes rule can be used to quantify a rational level of disbelief based on the observations.
This is convenient because it takes a lot of the what-ifs out of your question.  Could this happen?  Absolutely, without a shred of doubt.  Why am I so certain?  Because it happens all the time.  Ignoring the QM aspects, we are constantly coming up with beliefs about the world around us which are disproved eventually.  It's part of life.  Go take a journey through quantitative finance and the stock market, and you can see countless examples of theories which held true for a rather long time before being finally disproved.
This leaves only QM, which is now a relatively minor player in the game.  It becomes relatively easy to explore the probability that some particular set of extraordinary events can lead some from some reasonable priors to some faulty view of the universe.  This is true for any system which invokes statistics.  QM, weather, stock market.  Theories on all of these things point to the potential for unusual events.
However, it is hard to work with such things.  While such a region as you describe could occur, what do you do with that information?  Would you be able to develop better rational opinions?
There's one mainstream example of this thinking: quantum suicide.  This is a thought experiment akin to Schrödinger's cat, except instead of putting an innocent cat in the box, you put yourself in there.  You have a 50% chance of surviving the encounter.  But if you use the Many Worlds interpretation of QM, one of you survives.  You can repeat this many times, each time generating more worlds where you die, but at least one where you survive.  This is sometimes used as a test case for philosophies built around the Many Worlds Interpretation.
But what can you do with this information?  One quasi-scientific opinion is that, on the timeline where you survive, you are "exceptionally lucky," and that counts for something.  But it turns out to be very tricky to make this logic work out.
