Procedurally generated levels - check
Prema death - check
Pixels - check
Turn-based - false
Limited health - check


So anything similar to this is what I'm looking for - permadeath should be optional.

You could look at literally 99% of releases over the the last year here, and 95% of those are retro pixalated ones.

Try Dead Cells:

Perma death - check
Pixels - check
Turn-based - false
Limited health - check

and it's a fantastic Metroidvania worth checking out

The original Rogue doesn't quite fit these criteria. The way it works out here is as follows:
Procedurally generated levels - check
Prema death - check
Pixels - false (game uses ASCII characters instead)
Turn-based - check
Limited health - check, but your health regenerates

Specifically, is health regen, a feature common in traditional roguelikes, acceptable? (I should probably point out that food is limited, and monsters do respawn while you wait, so health regen doesn't make the game that easy.)

Off the top of my head:

Rogue Legacy
Ruin of the Reckless
Enter the Gungeon

What about something that uses less "turns" and more "loose periods of time, D&D style"?

Dwarf Fortress Adventure Mode:
Procedural generated world: Yes. Of varying sizes.
Permanent Death: In spades. You, your partners, and entire civilizations can crumble.
Pixels: Kinda, but there are tilesets.
Turn Based: Kinda. See the opening line.
Limited Health: Nope. Subsystem damage. Hypothetically you can lose all your limbs and still be ambulatory. (At a massive speed penalty.) Read more here!

Risk of rain

it pretty much ticks the boxes but you can regenerate health but if you are lucky to get health or regen items.

has characters to fit each play style and if you are lazy with unlocking them you can edit the save file to unlock all the dudes.

keyboard or controller, but I do say you get more control with a controller.

https://www.youtube.com/watch?v=kRhxe-OAsCk

Games that meet your requirements and have optional permadeath are unusual, so I split up my suggestions into games that do have it but lack something else, and the games don't have it but meet the rest of what you're asking.

Games without optional permadeath:

Nuclear Throne: https://www.gog.com/game/nuclear_throne
The best top-down, roguelite shooter.

Death Road to Canada: https://www.gog.com/game/death_road_to_canada
Randomly Generated Road Trip Simulator. with zombie hordes.

Enter the Gungeon: https://www.gog.com/game/enter_the_gungeon
Another top down shooter.

Spelunky: https://www.gog.com/game/spelunky
The paid for version has cartoon-style graphics, but the original freeware game has pixel art. You lose out on nothing by playing the freeware version.

Dead Cells: https://www.gog.com/game/dead_cells
2D hack-and-slash with persistent upgrades between deaths.

Barony: Cursed Edition: https://www.gog.com/game/barony_cursed_edition
First-person roguelike.

Games with optional permadeath:

Darkwood: https://www.gog.com/game/darkwood
Survival horror, but no pixel-art.

Tales of Maj'Eyal: https://www.gog.com/game/tales_of_majeyal
Roguelike with infinite lives option. Turn-based with simultaneous turns.

An upcoming game that could meet all your requirements and which you might want keep an eye on is Children of Morta: https://www.gog.com/game/children_of_morta

Dammit. Someone had to go and mention Dwarf Fortress and pique my interest again.

Syoban Action, a platformer famous for being rather unfair, has a "Mystery Dungeon" mode that might be worth looking into, provided you're not the type who gets frustrated when dying and you're OK if the game isn't always winnable. Specifically:

Procedurally generated levels - check
Prema death - false (you get infinite lives, death puts you at the last check point, and the level is re-generated at this point)
Pixels - check
Turn-based - false (it's a platformer)
Limited health - check (one hit kills you, and the game doesn't have any form of resource management)

I'm starting to think that maybe I should write a roguelike where you get infinite lives, don't lose anything on death (other than the level being re-generated when you die), and where the level generation favors random and chaos over fairness and guaranteed winnability.

Here is a GOGmix of "Randomized Permadeath Machine" games that might help you, although for most permadeath is not optional:
https://www.gog.com/mix/randomized_permadeath_machines_rpm

I got back into Super House of Dead Ninjas and was swiftly reminded why I hardly play these types of games.

Thought of a few others.

First, a serious one: Terraria
Procedurally generated levels - check
Perma death - optional
Pixels - check
Turn-based - false (platformer, again, but more of a metroidvania)
Limited health - check

Now, a silly one: Roguelike Simulator (https://eevee.itch.io/roguelike-simulator)
Procedurally generated levels - false (dungeon is always the same)
Prema death - check (though it doesn't really matter)
Pixels - false? (uses ASCII, though objects have non-ASCII graphics)
Turn-based - true
Limited health - false (death only happens when scripted; there's otherwise no resource management)

Finally, a romhack generator: Dragon Warrior Randomizer
Procedurally generated levels - overworld only; other areas not (though monsters and treasure are random)
Prema death - false (you lose half your money, or you can reload your save (but you can only save in one spot))
Pixels - check
Turn-based - check
Limited health - check
(Note that sometimes you may experience the sharp edge of the RNG; don't be surprised if a supposedly weak enemy casts a powerful spell on you, or if you learn said spell early.)

Bad news! There's been a major update. >:3

Here are my recommended Roguelike games:

FTL: Advanced Edition
Bionic Dues
Sword Of The Stars: The Pit: Osmium Edition