ZDoom

Is there any TC/MOD which inserts some East Asian culture into Doom?

Such a TC/MOD would likely take inspiration from Shadow Warrior, Legend of the Seven Paladins and maybe also Fate but anyway be unique.

Cultural elements of the Chinese, Japanese, Korean, Lao, Thai, Vietnamese, Malay etc. in level design, artifacts, weapons, enemies, etc.

There are a few, if you search for Japense Mythology ZDoom you will find some map and monster packs. One that looks god in particular is SEPPUKU, the thread has some links to other mods.

There's the Japanese Community Project; which, granted, is more Japanese-made and less Japanese-themed. But I had to mention it anyway, because it's pretty good.

There's Japanese Valentines which is the opposite, Japanese-themed but not Japanese-made. And bundled with HDoom for some reason; but if you ignore that it does have a few good looking maps. Especially the two by Pistoolkip.

Abysm 2: Spirit Falcon
Tarnsman's Projectile Hell
Touhou Doom