It's unfortunate that Hell Clock isn't on GOG as that looks like something I would play.

You could always message the devs to ask if they have any plans to put it on GOG.

I got it for Steam in the June 2016 HB Choice, the first game from that bundle that I've tried so far.

next one

the stanley parable?

actually, not quite. It is part of the Stanley Parable "universe" but not the "version" ppl mostly would expect

next one
I feel like I'm cheating with this one, but it's listed as a separate entry on IGDB, so I think it's fine

Dr. Langeskov, The Tiger, and The Terribly Cursed Emerald: A Whirlwind Heist?

no, maybe I should reinterpret the last hint:
- It's directly related to The Stanley Parable
- I feel like I'm cheating because it's technically a separate game, at least based on IGDB guidelines (so it's a separate entry there), but on the Steam store it's being put in a category you wouldn't expect it to be a full-fledged game
I'm starting to regret my decision to post this game lol

Is this The Beginner's Guide?

I know that's by the same developer as The Stanley Parable. Otherwise, I can't say how related they actually are beyond that.

I think I'll just make this session faster

I reckon in that case (and checking into the wikipedia entry) that you're maybe looking for The Stanley Parable Demo?

While i don't know the answer, the games on the wall are:
Gunpoint - shame that's not on GOG
Metal Gear Substance - assuming it's the PC version
Half-Life - FP shooters are very similar but this has a unmistakable "counter-strike" feel to it.
the last one, I swear I've seen the art style somewhere but can't quite grasp the game it belongs (screams double fine point and click)

Yes! It's The Stanley Parable Demo ...or rather Demonstration, both words are used interchangeably anyways.
Sorry for being overly specific, but I think everyone should try it, it's entirely a different game that tackles the topic of video game demos. Very wonderful "game".
It's listed as "standalone expansion" in IGDB, in the same fashion as Half-Life 2 Episode 1. Half-Life Uplink, the demo for Half-Life, is also listed as such in IGDB.
https://www.igdb.com/games/the-stanley-parable-demonstration

The demos displayed are:
Demo for Gunpoint
Demo for Metal Gear 2
Half-Life: Uplink - "Standalone" demo for Half-Life
Limits and Demonstrations - "Standalone" demo for Kentucky Route Zero

Stanley Parable demo, I was going to post that as my next game, was going to zoom into Half-Life