I'm so sorry about your favorite hen! That's so hard! You can rest easier knowing it was much more traumatic for you than for her.
If you're not going to eat or cook with the skin, you can hang them up on a tree by their feet, give them a post-mortem bath (if needed, I use dishsoap and scrub well, including legs, and spray off with a hose), and then skin like any game bird. Cut just through the skin, circle the legs, around where feathers meet leg scales just below the joint, make a vertical slice on the outside of each leg, then pull the skin away from the bird at the vertical cuts. Work the skin off like a glove, cut across the breast skin (with skin away from body), pull breast skin back towards the anus, and towards the head. Work the skin away from the back, cut the skin across the back and pull the back skin towards the tail and towards the head. work the skin off the wings. Once the skin goes as far as it can without the feathers popping through, I slice the primaries off the wingbone with a sharp knife and cut off and discard the third wing joint (wing tip).
You can cut off the tail here or later (if you're going to). I like to leave a flap of skin around/near the anus, sorta covering it, and that all gets cut off later when you pull the guts out and cut off the tail.
When you're done with skinning the body and wings, pull the skin down off the neck until you get to the head, cut the head off and discard along with the skin.
I do lots of rinses of the bird as I go, and try not to rinse the anus area after I start skinning to reduce contamination.
So basically you end up with a small flap of skin near the anus, and a skinned bird minus head and wing tips.
Remove from tree, place on table, cut off legs at joint where the scales start. Rotate, place bird on breast, remove crop from neck, cut crop and esophagus at body after smoothing them towards the crop to make sure most food is not present in the cord. Cut off neck. Rinse bird if needed. Rotate and flip over. Bird should be lying on back at this point. Cut abdomen just above anus to open the bird, widen this to both sides until you can get hand in easily, then reach in and grab heart, lungs, and pull everything out. Remove lungs if needed. Cut around skin of abdomen to remove anus connected to intestines. Also can cut off tail at this time as part of the area you remove in one piece. Remove edible entrails and save them (gizzard, heart, liver). Discard the rest of the entrails with skin and blood. Remove kidneys along the back bone if desired. Rinse bird well. Part bird now if desired.
Rest in fridge for 3 days or until rigor has passed. This resting can happen before or after freezing but if joints don't move freely prior to cooking, bird will be tough. Choose cooking method depending on age of bird. (anything over 6 months should be cooked low heat and slow methods, pressure cooking is most tender).
You can compost skin, blood, and innards, but must have a good method to keep pests and insects out of it.
If you look in the Sticky of this Forum there are some great training posts and videos, but not necessarily for skinning.
I haven't done game birds myself, but learned skinning from a hunting manual for game bird preparation and a butchering handbook, and from good advice on this site. CX skinning was super easy, my 1 yr rooster was more difficult due to more feathers and tougher skin ligaments. However, I pressure cooked the 1 yr rooster and he was delicious! Tasted like BBQ beef so I added BBQ sauce and had a sandwich. Broth made great pea or egg drop soup.
Hope I answered your questions. I think some folks remove entrails while bird is hung from the neck on a tree, but I never got the hang of that.