The same way switch works in every language and regardless of the clever pattern: find the first expression that equals and jump to that label.
Unless you use this pattern regularly - and most coders don't, even those who mostly write JS - you'll probably have to look the answer up. That alone is a good reason to stick to if/else; there's no such ambiguity there.
Also, by using non const expressions in case statements you lose the ability to have a large number of cases get mapped into a jump table. Although, I don't even know if more modern languages even support this optimization anymore since I think supporting non const case expressions is becoming the norm in a lot of languages.
Date parsing has different edge cases between browsers.
The DOM obviously provides more nondeterminism as well as deterministically different behavior, and there are some different things exposed to web pages so it's not like you can't detect which engine you're running on. But things like exact switch case execution ordering are well-covered by the spec.
Good point. By the way the answer is the latter, so the following prints 1 and 2.
switch (true) {
case (console.log(1), false):
case (console.log(2), true):
case (console.log(3), false):
break;
}