So, I actually need to bring along a Staryu/Eevee/whatever and evolve it? Or can I simply select "Use" in the menu for no actual effect in order to open it?
I believe it's the same as with the Escape Rope you need to use. You don't actually have to use it on a Pokemon.
For the Safari Zone when it means 10 days to wait for the next upgrade of blocks. Does that mean 10 days of using safari zone or just playing game? So to upgrade them all fully you would have to play 250 whole days to upgrade them all? or just a bit of every 250 days. like one hour a day?
You actually have to use The SZ once each day for the blocks/areas to upgrade. It doesn't matter how long, just as long as it's used that day.