I know I need each character's Passive Perception score in order to make secret checks to see if they discover hidden things by accident.
What else should I pin on my DM screen to speed up the mechanical processes of the game?
Different groups have different standards, but generally speaking, all you'll really need is their passive Perception and Defences. You'll be using those a lot, but you can easily ask for anything else you need in play. I've had GMs that ask for a full list of all skills, in case they need to roll secret checks of various kinds, but passive Perception and Defences is enough for almost all circumstances that I've encountered.
You typically use passive Perception for secret search checks, as you've said, and you typically use the Defences that you've written down to speed up combat. If you don't have to ask your players what their AC is every time you roll an attack, it makes combat go a lot faster.
For a similar reason, you might also consider keeping your players Spell Save DCs. If you can roll your monsters saves without asking each time, combat will go faster.