Q: What is a random outcome?

Any effect with the word "random."

Q: How does this interact with Court Jester?

You don't get to look at your opponent's hand, but you do get to choose which of their cards they discard (e.g. site or spell).

Q: If two are played, which one takes precedence?

The most recently played one.

Q: If there are two in play (and mine was played first), then I drop mine and pick it back up, do I take precedence?

No. Competing passive abilities are tracked based on when the card entered play.