This information was gained from the official site of Netherware Entertainment:
https://netherwareentertainment.com/
https://netherwareentertainment.com/pics/cheatcodes.jpg

1) You have to enter the UP-DOWN-UP-RIGHT-RIGHT-LEFT-ENTER combination in those couple of seconds when you see this logo on the start of the game (even before main menu appears):



If everything worked fine, you will see the new line ("Cheats") in the menu.



2) As it's said in the official picture, you can use only one cheat at a time.

3) You have to start a new game to make the selected cheat work.
That's a strange and unobvious thing, but this "Cheat mode" is not affecting the game that was already started. So, if you stuck on your walkthrough, you have to choose between "invulnerability", "unlimited ammo" and "unlimited saves", and then start a new run (because the cheat code will not affect the loaded game).

4) Yes, "Vanilla Flavour" is simply an invulnerability mode.

5) If you finish the game with a cheat code, the game will call you "Cheater" instead of giving you the rating at the final.

6) "The Gray Soldier" is a small additional mini-campaign that normally activates after the final titles of the game in case player gained A+ rating (which is hard to achieve). It introduces a new main character, and its mini-story-arc may (or may not) explain what happened to one of the protagonists of the main campaign after the ending.