Why do all inspectors in anime wear this stupid shit?
If its a proper trenchcoat? It keeps you warm and dry. That's two reasons which apply uniformly. A trenchcoat made as a piece of fashion wouldn't work, but a big wool trenchcoat will work perfectly, and a leather duster works when its warmer.
Outside of that, its hardly only anime inspectors. Earlier detective fiction has planted that image across the world as how a detective ought to look, with some variation based on culture.
Because Film Noir did it. Probably most famously with Humphrey Bogart as the P.I. Sam Spade in Maltese Falcon.
Also, it's who fedora faggots pretend to be when they wear their trilbies.
No. You might be if you go out and buy one without realizing it's a difficult outfit to pull off successfully. People looking like shit in them have basically ensured they'll be mocked everywhere.
Like fedoras, they look good if the person wearing it can pull it off. The guys in the Godfather wearing fedoras? Yeah, they look good. An obese teenager/young adult wearing it thinking it makes them classy and look good? Hell fucking no.
The problem with "noir" fashion nowadays is that people wear it without any consideration for how it sits or looks on them. With the right kind of face, and a suit, a fedora can still be respectable, but if you're a neckbeard throwing one on with a t-shirt and shorts you look like a tool.
Same thing for coats like this. They can look good, with the right ensemble and accenting features to compliment them, but a sweaty basement dweller will always look like a sweaty basement dweller.