The bracket has only been included in the box at the factory since November. Unfortunately, it's a bit of a lottery whether you get a cooler before or after November. However, you can have one sent to you free of charge from NZXT/support. Invoice of cooling and motherboard necessary. - Unfortunately, you then wait about 2 weeks until you get a bracket.

First of all, yes, the mounting brackets are supplied for Intel and Amd. I can't answer your question about the 12th Gen Intel exactly, or I don't want to give you any crap. In the other specifications you have "sockets" where you can see if it fits with your mainboard. If your size is not included, I would probably wait.