Ruin Explorer
Legend
The point I made as that they have similar styles of protagonists and similar levels of stakes.
What does that actually mean though, "similar styles of protagonists"? I guess I can't really discuss that unless you can define it a bit more. (I see you kind of did later - but if you want to add anything, I'd be interested).
Level of stakes makes sense, but let's look at it. The stakes in noir are pretty much never more than the lives of a few dozen people, and usually they're lives of a couple of people, and maybe not even whether they live or die, just whether they get rich or go to jail. I assume you agree with the above?
In S&S, the stakes can be as low as that, definitely - "The best thieves in Lankhmar", where spoilers, Fafhrd and the Grey Mouser think they're best thieves in Lankhmar but actually get ripped off by the real best thieves in Lankhmar, a lesbian couple who make fools of them. Very low stakes. But they can also be as high as the fate of entire cities, regions or perhaps even the world. Certainly Conan or Red Sonja have stopped some Very Bad people.
There's also a difference of stakes. Very often in noir, long-term happiness is at stake (and almost always not gained). That's sometimes the case in S&S, but a lot less often.
A S&S hero is jaded and mercenary because he is a strong man who only bothers to interact with society for the vices he desires.
A Noir hero is jaded and mercenary because society has broken him, torn him down and forced upon him vices that he cannot get rid of.
Well it's a little bit more complicated than that, but not a million miles away.
Your first description is bang-on for Conan so you're clearly learning something about S&S or revealing what you already knew but maybe didn't realize you did. There are definitely others like him, too.
But Elric, for example, kind of is mid-way between the two. He's from a society he's disgusted with and has thus left to wander the world, and he needs to the money to keep himself alive because he has physical problems that seem connected to that society's decadence (I actually forget the exact source).
And Fafhrd and the Grey Mouser love/hate Lankhmar like New Yorkers love/hate New York (and it is New York, as has been discussed), they don't not want to interact with it, in fact it's more like sometimes they tire of it, and have to go elsewhere, before inevitably returning. And initially they're not really jaded - they eventually become it, but it's a character failing that they correct.
Can you tell that Fafhrd and the Grey Mouser are my favourite S&S characters yet?

Re: your point on GH and internal consistency, I agree, but I think that just because something can happen and be consistent, doesn't mean it it's likely or typical. Certainly not saying that couldn't happen though. And whilst in noir it can often be difficult to distinguish good guys and bad guys (frequently the bad guy is themselves partly a victim of a setup from the dame/broad who kicks things off or whatever), in S&S, that difficulty is less common. It's more like evil badguys (often vile-ly evil) vs. morally grey heroes - many S&S characters are explicitly thieves, for example.