Hades

I can definitely recommend FTL: Faster Than Light. I still play it after many years in the exact same way you describe, that is short sessions. t's all based on battles that last minutes at most while part of an overall campaign. You can quit anytime, even during battle, and it will just saw the state. You can pause during battle to think as well.

it also has simple rules but with a lot of depth. You upgrade your ship from time to time and also get new weapons or defences. It actually sounds like a perfect fit for what you want. It's cheap too.

Slay the Spire ! If you like card game and roguelike

Slay the Spire

Hollow Knight is a masterpiece worth recommending. Furthermore, it satisfies most of your criteria. But you might need one of the better integrated GPUs to run it smoothly on your system.

Hades is another masterpiece worth recommending and perhaps satisfies even more of your criteria. Though, once again, it requires you to have one of the better integrated GPUs.

Factorio.

There is a free demo, start there. Most people can get a good 20 hours out of that if they enjoy it.

It can get somewhat complex if you go with mods etc. But you will get 100 hours out of it before you go down that path

The full game is worth every cent (5500 hours later...)

Dead Cells is a game I always have installed just to pick it up in bursts of 30 minutes or an hour.

It's a roguelike, it's challenging and it's easy to pick up any time.

Even though it has levels, the intended way to play it is in runs. You start the game, start a new run, and try to go as far as you can, you die and repeat.

Multiple paths to choose, so it never becomes boring, and the levels are generated, so you can't memorize everything.

The Binding of Isaac: Rebirth sounds like what you’re looking for.

Vampire Survivors (and just get the DLCs as well, it's cheap)

You'll never get more bang for your buck on steam. 60 hours and counting, even with only one DLC. I just keep unlocking amazing new stuff

Any of the Portal Series. You can slowly progress through the game (first one ain’t long at all) and it has a great story.

3 different genres:

• Serious Sam
• Running with rifles
• Baba is you

Sounds like you'd enjoy Nuclear Throne or Enter the Gungeon. Both are very similar, they are roguelike in 2D top-down view with enemies that shoot a lot of projectiles and gameplay is about using dodge rolls and taking cover

Have you tried Double Action Boogaloo? It is probably right up your alley if you like Fistful of Frags.

You might enjoy Quake or other arena shooters like Warsow or Ratz Instagib.

If you want to try something new that still fits the bill I'd go with Enter the Gungeon or Dead Cells. Both are easy to pick up but incredibly challenging to master. Both are roguelikes with a short time per run, so the game already does the job of breaking itself into short play periods. And both are action heavy.

HoloCure. It's on itch.io and Steam. For free.

• Assault Android Cactus
• Electronic Super Joy
• Jamestown
• Lifeslide
• Flatout 1&2
• Splatter - Zombiecalypse Now
• Yoku's Island Express

Dicey Dungeons, Streets Of Rogue and Enter The Gungeon

From the top of my head, all these work perfectly on my 5yo Intel laptop and are often found on sale or in bundles.

Baba is You: you know that in every game, there are a fixed set of rules ("physics"), and you must use them efficiently in order to win? In this game, you must change the rules to solve puzzles. Super simple gameplay, tricky to master, really fucks your brain as you need to think outside the box.

Hotline Miami (1 and 2): top-down shooter with impeccable gameplay, level design and soundtrack. Super fast paced, die-retry-die-retry game loop, and great story too. Every level is challenging in its own way which makes it not so repetitive.

Nuclear Throne or Enter the Gungeon: procedurally generated top-down shooters, very similar to each other. Fun pixel-art, never replay the same levels although I guess it could be repetitive after a while.

You might enjoy Broforce. Very chaotic and fast.

Another force-based recommendation is Dustforce, although that one might be a bit repetitive.

Not a Hero is pretty frantic and levels are over very quickly.

Risk of Rain 1 has native Linux. 2 does not but runs on Proton.

Check out my game DR4X :D Its a fast paced, and simple 4x game where you can finish a whole game in less than an hour. You can also fully customize the setting of your game, down to like, the types of monsters and bandits and random events and tilesets, etc. https://store.steampowered.com/app/1704830/DR4X/

Its also only 5 bucks (it's on sale for the steam strategy fest)!

Super smash brothers melee

If you want to pay: Anything by Adamvision Studios (Note: Force compatibility mode for these games on Steam, the linux versions are wonky) Demon's Tilt or Xenotilt

Free: Neverball (this one is probably fits best what you're most looking for) avail in Software store on most linux distros

A few games that I think could fit your requirements: Just Shapes and Beats, Among Us (yes seriously), Team Fortress 2, Rocket League

I play Tetravex - the 4 x 4 version. It's available for free in the games section of Linux Mint or Gnome.
There's also a free android game on the Play Store.

I'd recommend Borderlands 2. It runs beautifully even on Intel HD4000 and it's less than $5 on sale. It's much better with friends, but I've enjoyed most of it by myself and absolutely love it.

You may also try Hero's hour and Death road to Canada.