It says the King's Court cards are supposed to be face up the entire game, but when you get a card that says to pick one, it says "random." How do you pick a random card when you can see them all??

You could take them all into your hand, look away, shuffle them, and then have you or another player pick from them (revealing only the top edge perhaps).

Or you could make use of the die. A d10 can be split four ways evenly:

1/2, 3/4, 5/6, 7/8, reroll/reroll

or

Roll twice:
Card one: Odd then Odd
Card two: Odd then Even
Card three: Even then Odd
Card four: Odd then Odd

When fewer than four are available you'll likewise have to improvise. I actually added a d8 and a d6 to my set to help me randomly pick monsters of the same type. When you have a d6, d8, and d10, you have the ability to randomly select between 2, 3, 4, 5, or 6 different things without having to have a "re-roll" number.