A coop PvE game for two dads and three sons?

Minecraft! There's lots of modpacks as well that give it variance and replayability

Minetest! FOSS client and server 👍
Especially now with the expansion of “datapacks”. I hate to say it, but Bedrock version of Minecraft might be the way to go. Java would be fine too, but unless they’re a bit more technical it can be harder to get multiplayer going for mods.
- I did this previously by using MultiMC (now PrismLauncher). Make a mix of mods, send them the zip of the whole game instance and ask them to drag and drop it onto MultiMC. The biggest issue I encountered was one family member having a black ingame world until they changed a setting in the graphics mod, otherwise it didn't seem to be too hard.
  
  By comparison Minetest is much easier for playing mods with family. Everyone downloads the server's mods when they join. But the interest is lower.

Terrarria has 2d shooting, cartoon violence that is kid friendly, and you can have up to 8 players in a world.

Palworld, rent a dedicated server, anyone can hop in or out as necessary. Guns, legally(?) distinct from pokemon monsters to catch. Cartoon violence. Tons of fun to be had.

Not FPS, but Valheim is plenty fun. Also, zomboid, but that ones a bit more violent

Valheim is also capable of being extremely frustrating, so I'd recommend one of the mods that removes the skill penalty loss when dying. It might not be necessary, but dad knows the kids' temperaments better than me.
- EpicLoot might also be good, if OP's kids are keen to grind components.

A survival game is probably what you’re aiming for here. Someone has to host in those scenarios usually unless one of you pays for a dedicated server. I don’t have a ton of experience with the shooty kind. But Valheim, Vrising and Palworld were a lot of fun for about 20 to 30 hours each. Palworld had some shooter experiences.

I like these because it usually turns into a “let’s get things done together “ and everyone finds their tasks to do. Then you put it all together and you end up with a cool base you built together and have some pride in. Palworld again is the one with the lowest “violence” imo

I would definitely recommend some roguelike shooters like Roboquest, Gunfire Reborn, Crab Champions, etc. These are all 4 players, but you could always have a few different lobbies.

Maybe also consider something like Palworld, Grounded, Raft, and other Survivalcraft games. There should be a good few in that genre that will allow you all to play together!

Sounds like a fun group ready for some great memories! Good luck on your game hunt!

Man I wish roboquest supported created than 2 player co-op. That'd be so damn fun
- I've played so much of it and didn't realize it only supported two players! I guess I'm glad I only talked one friend into playing with me 🤭.
Perhaps Valheim - the archery is pretty good and while not as twitchy as a shooter might suffice to scratch that itch
- There's always the azumatt first person mode and Better Archery mods to get you there 😊
I believe Grounded is limited to 4 players too, unfortunately.

I'd like to recommend Crab Champions. It's currently discounted, and quite a blast. Sadly, it's only four players simultaneously, but quick and also fun to watch.

A shooter might be hard, but I think you could make ORION: Prelude fit. It's a little old now, but still fun. You get to shoot dinosaurs and run around big maps.

Other games that I'd recommend: Avorion (build your own spaceships and fight and befriend the galaxy) It's not technically a shooter, but depending on whether you pick out lasers, electric tasers, machine guns, or rail guns, there is definitely some aspects of moving to avoid enemy fire while plunking them down. [but don't be like me and just make lots of borg cubes... because cubes are the best!]

Ark: survival evolved (tame dinosaurs and fight the map)

Elden Ring with seamless Coop mod (up to 6 players at once)

War Thunder for plane sim, World of Warships for good ship battles

It's not a shooter, but Enshrouded is insanely fun and you can play with up to 16 people at once.

Don't Starve Together - not a shooter though.

CoOp / PvE + Shooter + 5 players is a bit of a challenge to be honest...

Is one category less important than the others? Might help with suggestions.

My first thought was Borderlands 2, but 10 might be a bit young and it doesn't really qualify as "as little violence as possible".

Maybe Deep Rock Galactic, or Vermintide 2? (Or Darktide but that's far more violent and it's humans you're killing.)

Earth Defense Force might be fun, too. It's very silly, the humor would probably land with a 10 year old.

Risk of Rain 2, maybe, also.

He asked for a 5 player coop game and you listed exactly 5 games that are 4 player max.

You are awarded no points, and may god have mercy on your soul.
- There are (modded) DRG lobbies with extreme player counts, and I don't think it's that hard to set them up. The problem I see with DRG is that it can be pretty spooky at times (similar to Minecraft or Subnautica), so it might not be suitable for a 10 year old.
- Others have already mentioned RoR and DRG; VT2 and Borderlands also have mods that support at least 8 and I think up to 16 players.
  
  I think EDF is the only one that definitely doesn't; I was mis-remembering how many can play that, I guess. Thought it was 5 or 6.
- You can mod risk of rain for many more than 4 players.

There are a lot of great 4 player games out there.

No love for five....

Helldivers 2. It's not violence when it's justice, citizen.

Not 5 player coop though

If you can find it, Evolve. Four players fighting the fifth player as a monster.

Good suggestion, there are a few in this genre. Dead by Daylight and Friday the 13th are two others. They're made for 5 people, as long as you're ok with one person being the monster.

SvenEvils has a great list of local co op games. https://store.steampowered.com/curator/8485829-SvenEvils-Playground/

Ninja Turtles Shredders Revenge is up to 6 players. Technically it's tons of violence, but it's cartoony. You can do online or couch coop too.

@janbartosik @gaming Bronzebeard's Tavern is free, and 1-8 player:
https://store.steampowered.com/app/2422870/Bronzebeards_Tavern/

Not a shooter, but you can have a lot of fun in custom matches of Rocket League and the game is suitable for kids and free.

@janbartosik @gaming Helldivers

Satisfactory could be amazing if everyone's into that sort of thing -- there's enough to the exploration and pve that players don't have to just be building factories all the time, but if they do get into the factory part it's an excellent foundation for getting into engineering one day!

DRG can be modded to support 5+.

I know you asked for as little violence as possible, but have you considered Destiny 2?

There are single, two, three and six person activities (and you can use group finders to help find a 6th and maybe make a friend on the way).

It is a first person looter shooter, but it also has a deep story as well as mechanics (eg platforming, solving puzzles, exploring areas to find collectibles). Ultimately though you do need to shoot and kill stuff. It's not that gory however.

There's also builds in the game, so you can spend a lot of time tweaking your builds to get it how you like it, aka theory crafting.

It's also "free to play", but in reality it's "free for the first bit and then if you want access to heaps of other content you gotta spend money".

Destiny 2 is a terrible choice for a 5 person group, the only activities are raids which is endgame

Helldivers 2! It's not violence if it's bugs