the free diamong thing, when you buy diamonds using the starter pack when starting a new city, you get 7 days of free diamonds.
They go into your story quest section to collect each day.
If you have Chateau Frontenac they multiply according to the level of your CF.
So 500 diamonds a day becomes 1000 a day for 7 days at just level 6 CF and so on.
If you buy more diamond packs then they sit in your inventory, you don't need to action them until your ready, so you could wait until you really needed diamonds and you had a chance to level up your CF to make it even more, 150%, 200% etc