Question 1: How exactly does time pass in ETS2?
If you go to your garage (usually at certain gas stations too), you can stop your engine and sleep for around half a day. However, you can only do it 3 times at once before it removes the option for a little while.
If you have 3 trucks and 2 other drivers, go into the Truck Manager screen and keep switching trucks with the other drivers. When you switch, it should say "It'll take X hours to switch to this truck." or something similar. Keep doing this until both the trucks are "Idle" or enough time has passed.
Playing the Game
Pretty simple. Just go around doing jobs and driving around.
Question 2: Does the method how it passes affect delivery offers too?
Yes, anything that makes time pass affects delivery offers.
Question 3: Does it pass when you are chilling in the menu?
No, the menus pause the game completely.
Hope this helped.