Don't know the exact criteria but according to Papers, Please Wiki:

• Class-8 – This is the initial apartment provided to the inspector and his family at the beginning of the game.
• Class-7 – Available at the end of day 12 at the earliest.
• Class-6 – Available at the end of day 17 at the earliest, if already living in a class-7 apartment.
• Class-5 – Available at the end of day 22 at the earliest, if already living in a class-6 apartment.

Also, according to this Papers, Please Wiki discussion thread there might be faster ways, but for now, this is the earliest proved.

Disclaimer: I do not know the Code of the Game

It is probably randomized. At least what I could gather from information. There will be a check if a day has been reached, and probably how much money. Something along the lines if(day >= 6 && money >= 1000) and then everyday there will be certain chance that the script for the offer will be run.

I guess that this is the case, because a lot of papers please is randomized. There are many events that happen at random and even though you can force certain events to happen it is hard to get event X happening at day Y. Usually you can force event X to happen at all and it will happen at day Y+/- 2 days or so.

But again this is just a guesstimate and I do not know the code. I know this answer might not be very satisfactory to you, still hope it kinda helps.