I ground my pack 2-3 yards away from the tree, on the face I'm going to climb on. Tie it to the end of my rappel rope. Set my weapon on my pack so it's out of the snow, mud, etc., and attach the carabiner on my haul line/pulldown line/gear hoist/dog leash. Being yards away from the tree, any branches or bark or snow I kick loose while climbing won't fall on my gear; added bonus if I fall I'm less likely to land on my own broadheads. At height I haul up my bow/rifle and hang it, switch my haul line to my scaffold knot so it's now my pulldown line, then haul up my rappel line with my pack at the end. Before I hang my bow I do a couple of practice draws, both to function-check the bow and to identify any twigs or branches I need to trim. With a rifle I ensure I have a round chambered, safety on and clear scope picture, or hammer down and unobstructed peep sights.
At the end of the hunt I'll either just drop my pack to the ground, or lower it on a bight of my rappel line. I generally try to drop it away from the trunk so I don't land on it. Lower my bow on the end of my rappel line; I try to 'fish' it so it comes to rest on my pack. Rappel down to retrieve my 1-stick or ROS or platform; use the attachment strap or rope to sling it. I used to just toss it down but would invariably catch my aider on a branch and have to wrestle it down. I've got 40' of rappel line so I have plenty-enough slack that I won't jostle my bow while rappelling.
Good thought on the pillowcase. When I helped with the hunter safety program, the COs always suggested including a soft gun or bow case in your pack; if you have to hitchike or whatever back to your ride you can case your weapon and stay in compliance with DNR regs. I'm suddenly picturing a large-ish rectangle of blaze orange fabric that can be zipped or buttoned to a DNR-compliant bow case, or use as a signal panel in an emergency, or any one of a dozen other purposes a clever marketing person could come up with.