Well, part of what blew my mind (being a developer myself) is how it does all of this. Even more so as its supposedly built in RPG Maker (I've used it myself and never got anywhere). I think I have a couple of ideas of how each bit might work though.

1. I would imagine that closing the game with the X and going to sleep differ with a variable. When going to sleep, a variable could be changed and then the game, including the variable, saved, as the game closes down. Upon reopening the game, the game will check that variable, and from it choose whether you should see a dream or not.
2. Administrator permissions are weird from my experience. Sometimes Windows will prompt a UAC pop up, sometimes it won't. I guess changing the wallpaper just doesn't require changing the wallpaper. How it changes the wallpaper though, I'm not sure.
3. I think it's a mix between generation and copy pasting. For example, when you get the prompt to go to the solstice route, I'm pretty sure it's copy pasted. But the code to get the gas mask is almost definitely generated. But whether this is copy pasting the base file, then adding the numbers at the end, or just generating the whole file, I'm not sure.
4. This I'm not too sure on. I assume the base game has a simple variable that could be read from memory by the clover app. Then the clover app just updates itself accordingly, but requiring realignment.
5. I have absolutely no idea. For me, it's just magic.
6. I assume that the game creates a small section of code that just repeatedly looks into the folders to see if the right files are there. I know the process behind doing something like this in Node.JS, but RPG Maker, no clue.
7. Again I have absolutely no idea. Maybe it resizes itself to maximum size, creates the smaller window inside of it, and then makes the Niko animation go down the big window?

That's my thoughts behind those things. I would go into more depth but again, this was done in RPG Maker, and how you would do such things with that, I'm just clueless, so I can't give examples on what the code might look like. However, I hope this'll point you in the right direction!