Deep Rock Galactic fulfills nearly all your requirements, depending on A) how it runs on your machine, seems Linux works for many and B) if alien bugs count as zombie for you, as they are also mindless hordes that swarm you. I have a lot of fun with DRG, I can recommend it.

Trine Series contain fun coop puzzler for up to 3 people. Don’t know how all of them play on Linux, but 1 & 2 work on the Steam Deck, so should work in general. And 2 is in my opinion the best one. They have a story, but its not really about it, the puzzle and levels are 95% of the focus.

Lara Croft and Tempel of Osiris is up to 4 coop puzzle action adventure. It was fun and the difficulty 9f the puzzles gets adjusted for the player number if I remember correctly. Seems for some it runs on Wine, but as always, mileage may vary.