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#1
if that is a 3rd party cooler there should be instructions to mount it. Most would come with a backplate to screw into. If it is the stock Intel cooler, the plastic 'screws' in the last picture look about right. They just push through the motherboard and hold the cooler to the board.
You need the correct backplate. I like the screw-and-backplate mounting over the push-pins!
To keep from damaging your CPU (or cooler) you need to follow the instructions supplied with said cooler.
And since we don't know the CPU cooler you have, Guessworks here can have disastrous results for you.
Both the CPU and the Heatsink are from an Acer Nitro 50 (the case was too small). The MoBo had those little cylinders to receive the screws (studs?)
You answered one part of the question but you didn't address the most important one - instructions?
You need a backplane like this:
Amazon.com
Some CPU coolers come with a backplane:
https://www.amazon.com/Thermaltake-7...dp/B002RWJGI6/