Minimum dimensions of a mini-ITX case for a stock Intel cooler?

Question

Frame

I want to assemble a compact PC on the mother of the mini-ITX form factor for the LGA1155 socket.
I chose the motherboard and the processor, but ran into cases, the dimensions of which are very different.
I want to take a flat case, vertical is better, but horizontal is also possible, if it is good.
It is not supposed to install expansion boards, I want the minimum thickness, so I measure it by the processor cooler.
1. Tell me, are there any standards for the internal thickness (height) of the case, so that i3 / i5 fits into it with a standard BOX cooler? What is the minimum?
2. If you know the models of flat cases that these coolers fit exactly into but with the minimum allowable gap, please advise.
3. Overclocking is not expected (possibly downclocking), a little noise is not terrible, so I don’t want to change the regular cooler for no reason. Or are there good low-profile coolers at a low price?
Previously laid eyes on:
ASUS P8H67-I Deluxe
i3-2100T / i5-2500T

Answer 1

look at the cooler itself, what kind of slim can be here
. There are standards. see form factor
There are non-standard ones, but this is rare and may not be compatible with anything.