Whether: It's a yes-no question word-- "Whether he's right or wrong, it's still important to respect him as a person."
Among and Between: Simply put, "between" is for two things, and "among" is for several things. So you might say "They fought between themselves over whether or not to go the the party". That means you know there's only two people. If they "fought among themselves", you know there's more than two.
There is an exception, though. Sometimes when several things hugely contrast with each other, you won't use "among", because "among" sounds too much like they're *one group*. That's when you might say "I have to choose between the red car, the blue car, and the black car." In this case, saying "choosing among" these three cars makes them sound like you're choosing randomly, as if you're picking from a box of chocolates without thinking.