As a note, if you stabled your horse in one stable it isn't automatically available in other stables, you have to go to that specific one to get it. Like the garages in GTA.
I found this link helpful. With a varmint rifle with one shot to the head it should give you a perfect pelt.
https://www.gamerevolution.com/guides/450389-red-dead-redemption-2-get-perfect-pelts#/slide/1
I too would like that from an immersion standpoint but as smoking is designed to refill certain cores they likely shortened it for people who use it as a tool, not for immersion.
Yes in the stables. Not sure how far into the story you are but if you haven't visited the stable in Valentine yet then you can't rename your horse yet.
"Also revealed on the box in fine white print is the fact that Red Dead Online will support between 2 and 32 players."
https://rockstarintel.com/ps4-users-to-get-exclusive-rdr2-online-content-30-days-before-xbox-users/
You do keep them both, but if you replace one it will go back to your horse. If you want to hold both you need to select in your weapon wheel either an empty slot or a slot with a weapon you don't mind putting back onto your horse.