Horror and puzzle eh?
Downfall gets my first recommendation. It's a point & click horror game that is darker than most games, in both theme(s) and visuals! Now there is a remake of it currently in production (to be released later this year) so the publisher has decided to offer it for free from their website, or you can buy it here of course. If you want it as a freebie just visit the publisher's site and click on Free Games
http://screen7.co.uk/
Next up is The Cat Lady. The Cat Lady is a very loose sequel to Downfall and in my opinion is better than it, and better than most games here. It's dark, distributing and occasionally...uplifting? Yep, a game that starts off with the main character narrating their suicide note somehow finds a way to have some positively uplifting moments. Can't recommend this game enough, go grab it!
As for FPS and horror, the first that come to mind are F.E.A.R. and F.E.A.R. 2. I've only played the first F.E.A.R. but I found the AI to be outstanding (when it has room to move) but the rest of the game was just so so. The horror bits were (for me) predictable and well...just not scary.
But fear not! (pun mildly intended), there is Nosferatu: The Wrath of Malachi, a horror FPS with roguelike tendencies. Pretty simple game mechanics, but tons of fright to be found. There are occasional jump scares but that isn't the main source of horror, rather it's the heavy handed and excellent atmosphere, complimented by music that will keep you on the very edge of your seat! Even when you expect to find an enemy you'll still jump a bit, thanks to the wonderful air of suspense.
If it's horror alone you want, but in a 3D world then check out the Penumbra series. You've probably heard of the Amnesia series, yeah? Well Penumbra was created by the Amnesia devs before Amnesia was a thing, Penumbra is basically Amnesia V.1, and in my opinion better thanks to more likable characters and stronger writing. I would also venture forth that Penumbra had a better atmosphere than Amnesia, few games have made me feel truly as "alone" as the first two Penumbra games.