The way the MWI treats probability is roughly as follows. You take the non-probabilistic part of quantum theory. You then assign decision theoretic payoff to each possible measurement outcome and assign a value to a state that respects non-probabilistic axioms of decision theory. One example of such an axiom would be that if two states $|\psi_1\rangle,|\psi_2\rangle$ are the same except that every outcome has an increased payoff in $|\psi_2\rangle$ then the payoff for $|\psi_2\rangle$ is higher than that of $|\psi_1\rangle$. David Deutsch has explained that you can derive the standard probability rule given those assumptions. The MWI can be tested using those probabilistic predictions. If a particular event would make your theory problematic then an experimental observation of that event is a criticism of your theory.
Now you ask about branches of the multiverse in which extreme events happen. Note that if this is a problem, then it is also a problem for theories in which there is a single universe and stuff happens randomly since unlikely stuff can happen in such theories. The main difference between the two cases with respect to probability is that the MWI explains the probability rule and the circumstances under which it is applicable, and the single universe theory postulates it arbitrarily. So any refutation of the MWI on that basis would refute all probabilistic theories. But in both cases, this amounts to saying that a measurement can go wrong, which can happen for other reasons. If you make such an observation, then you should check out a load of other sources of error before you consider the idea that you're in a very improbable branch of the multiverse.
Another part of your question is about whether we might see the wrong laws of physics, e.g. - the second law isn't really true. The problem is that this treats the second law as being an arbitrary postulate that could just as easily be completely different. But that is not the case, the second law is tied deeply to other parts of physics like the theory of information. See, for example, "Maxwell's Demon 2 Entropy, Classical and Quantum Information, Computing" edited by Leff and Rex. Replacing the second law requires replacing a whole load of other fundamental physics.
