But isn't it up to the legislative branch to make/amend laws? Why elect the judiciary branch, which is only supposed to interpret those laws?

This is actually a big deal in America and a fundamental change in how our government operates. Your idea is right, but since the 1960s judicial authority has grown in a way that increases their responsibility to the point where they're almost legislators themselves.

Because there's a lot of different ways to interpret laws, judges have a fair amount of flexibility in sentencing, and elected judges aren't by default just cronies of whoever appointed them.

I think prosecutors fall under the executive, not the judicial, branch

I believe that's the idea, yes. However it's very easy to see that it doesn't actually work that way. Unseating an incumbent candidate in local governments is damn near impossible, barring "scandals" or obvious extremes (but not even then, depending on the area we're talking about).