where does the little black star go on the hat?

It goes on the hat, I bought one back in high school (graduated 2012) and it came on the front of the hat like in the pic.

Does the hat come with the soviet star affixed, or is it a separate pin?

Not sewn on, but its a pin that is pre-attached to the hat. to remove, look behind the flap from the top, unbend the pins and slide the main pin out from the hat

Is this genuine soviet surplus or a reproduction?

I want to say this hat is a reproduction.

How do i put on the badge

In installing the badge, it might be necessary to punch a hole into the fabric using the pin of the badge. I do not like it because the hat is so beautiful to punch a hole in it. So I decided not to install the badge. But if I need to have a hat with a badge on it, I buy another hat for that purpose. Therefore I actually have two hats. I plan to buy more each with a different badge. To minimize damage I used a small needle to punch a hole at the front of the hat instead of using the pin of the hat which is quite big.