It's no worse than a single world interpretation: These things happen all the time, the branch of math that deals with them is called Large Deviation Theory (and it's closely related to information theory).

One of the corollaries/interpretations of Sanov's theorem is that, generally speaking, when faced with an astonishingly improbable outcome (e.g. flipping 9,000 heads and 1.000 tails out of 10,000 independent coin flips), no statistical test can differentiate between "that improbable occurence with a fair coin" and "an unfair coin" - the fair coin, when it does something improbable, with have (with overwhelming probability) a specific tilted distribution that looks unfair.

But in single interpretation only one outcome happens on a trial. The full distribution does not manifest on a single draw. In MWI all the possibilities occur, which is different

Somebody usually wins the lottery. That doesn't change the fact that it's really unlikely for any given player to win the jackpot.

I guess you win. MWI is really not different from classical probability in any way.

Any state is equally improbable. It's human judgement to tell something was special or not. If all states are all possible combination of numbers in a lottery, you just call 1 combination "I won" and all the others "I lost". Same thing applies to a cleaned up room, any arrangements of objects in it are equally probable, but the number of states you would call cleaned up are so much less than the messy ones, so you can say a cleanup up room is less likely than a cleaned up one (unless you do something about it :P) This is also the fundamental of entropy btw.

TL;DR By grouping states together (human choice), certain arrangements seem more probable than others.