How can I set these nails flush in my beadboard paneling?
Unless I've got the scale completely wrong, those aren't finish nails. These are finish nails, since they have tiny heads. A finish head nail is used when there is no force on the head, such as trim and other decorative pieces.



What you look to have are common head nails. These are used when the force on the nail head is significant and avoids pulling the nail through the wood.



You can still countersink these, but not with the nail punch you have, which is designed to completely cover the nail head so it won't slip off. You'll need a ballpeen hammer to get that done. You use the rounded end to pound in the nail. This is done either by using it like a regular hammer and risk missing the nail, or you put the hammer on the nail and use another hammer to hit it.



And no, hitting a hammer with another hammer isn't going to make either explode. This was busted by the Mythbusters years ago. You can break chips off the hammer, if it's old and already showing signs of stress, but it's not going to explode.

https://en.wikipedia.org/wiki/MythBusters_(2006_season)#Hammer_vs._Hammer