My team uses airbnb's styleguide as our default for coding style, although we do deviate occasionally.

Does airbnb have an opinion on where the operators should go in a multi-line ternary?

It's not specifically talked about anywhere, but there is the example:

// better
const foo = maybe1 > maybe2
  ? 'bar'
  : maybeNull;

one of our engs prefers this, with the justification that it's easier to read. If you read the first line by itself, then having the ? at the end lets you know that this is a ternary and is continuing. Otherwise, if you're skimming the code very fast, you might think that you're assigning foo a boolean value.

const foo = maybe1 > maybe2 ?
  'bar' :
  maybeNull;

And again, we're using multi-line ternaries because we have some long expressions and values to be assigned.