I keep running across references to letting the meat rest, but no detail on why, for how long, or whether this "resting" needs to be done prior to or after gutting.
I'll go through the three things.
Aging is when you keep the bird long enough to let rigor mortis pass. If you cook the bird immediately (cook, not just butcher) rigor does not have time to set up. As Cassie mentioned, Mom would cook the bird immediately, no aging. no problems. You can tell when rigor has passed by the meat. If you have joints the joints should be really loose. With no joints the meat itself is very loose and limber, not stiff at all.
Brining is when you soak the meat in a salt and water solution, often when aging. There is some debate on whether the salt actually tenderizes the meat, I don't think it does directly but others do. What the salt does is retain moisture in the meat. If you are going to cook the meat a dry method like grilling or frying brining can help. If you cook it a wet method I don't see a lot of benefit in brining. You can add a salty taste at any part of the process.
Marinading is when you keep the meat in an acid solution. The marinade is usually wine, vinegar, or tomato based. The acid breaks down fiber in the meat. The stronger the acid and the longer it is left in the marinade the more the meat fiber breaks down. I don't consider a marinade necessary for really young chicken, might even turn it mushy, but for an older chicken it can really help. An example is Coq au Vin, how the French turn a tough old rooster into a gourmet meal.
You can add flavor at any stage. A lot of people use a marinade for that. The flavor itself doesn't affect tenderness but the acid will.
when you buy a chicken from the grocery store, I was taught it needs to be cooked within 24 hours or frozen right away. Does the freshly defeathered chicken not begin to stink/rot if left to rest in the fridge for a few days?
The others have covered this. The devil is in the details. The fresh meat you buy (poultry, beef, pork) has been handled and treated certain ways. That's going to include aging. You don't know how long it has been aged. They also don't know how you are going to store it. So they give the advice to cook it immediately, to a large extent because they don't trust the customer to know how to handle it.
That wasn’t a very safe procedure! As mentioned before, the innards are the first thing that breaks down, I wouldn’t advies ever leaving the innards in, I pluck the chickens and then get the innards out , after that, you wash the meat thoroughly and put it in a fridge
I also like to get the innards out when I butcher and not wait. I just feel better about it. I'm making a mess while butchering, make one mess and get it over with. Again, the devil is in the details. If you study how they hang pheasant in Scotland they often leave the innards in for several days. The detail there is that it needs to be cool enough that the bacteria doesn't grow. I don't know the details in their techniques so I just clean them out. As mentioned in that article, a larger bird takes longer to cool down.
There is a reason hog killing time was in the fall after it turned colder. After Dad got a day job he did not have time to butcher our hogs himself. The country butcher he took it to had a huge walk-in cooler so he could age the meat, even in warmer weather.
You probably gathered this from your video, but you don't want to leave your bird in the hot water very long for the defeathering process. Certainly not long enough that the bird starts to cook at all. Just enough so the skin relaxes. Good luck to you!
If your dunking water is too hot or you leave the bird in the hot water too long the skin can tear when you pluck. That doesn't make plucking harder but makes for an ugly carcass. If you cook with the skin on and appearance is important to you that can be a problem. Personally I skin mine instead of pluck, my wife wants it skinless anyway, so I don't bother heating water to pluck.
We will likely burn the blood and any bits (not the feathers) we don't plan on utilizing. Does that sound ok or would you change anything? What do you all do with the feathers, if anything, besides discarding them?
We all have our own ways of doing these things. Whether you are urban, suburban, or rural can affect your options.
When I butcher I have two buckets. One is for the bits and pieces that I'm going to feed back to the flock immediately. I only save enough that they can finish it before night, I don't want the leftovers attracting predators though I don't worry about any blood left on the ground. The other bucket, which includes feathers, gets buried in my garden in an area that won't be disturbed for a while. The garden is fenced so dogs, coyotes and such can't dig it up. I used to bury it in my orchard. There I'd cover the area with wire weighted down with pavers so critters could not dig it up.
If you are going to put the offal in your trash you might want to freeze it so it doesn't stink before pick-up. Some garbage pick-ups don't take the leftovers from butchering.