1.) How do you chop off the head while the deer is hanging? (Sawzall?)
I use the sawzall. Bow saw will work too. Use a knife to cut through the hide and muscle all the way to the bone, then use the saw.

2.) Do you remove the innards with the deer still hanging vertically by the hind legs, or do you use the FEL to change the working angle on the deer?
Yes, I hang the deer from a loft floor joist.

3.) We just leave the innards on the ground for the coyotes, but I hate coyotes. (It sounds like an idiotic practice when I type it out.) Should I make a little burn pit in a safe spot and get rid of the innards without attracting coyotes? Do deer hate wood smoke? (I don't want to drive off deer when I have people going out again to hunt the next day.) Do deer hate smoke with burning innards smell included?
I don't like putting gut piles out where the varmints can get them. We build a good fire in a 38"x16" tractor wheel rim and burn the offal and bones. Never noticed it bothering the local herd.

4.) We usually don't skin, because ground skinning (even with a tarp) gets dirty meat when the wind starts blowing. They charge $10 or $15 to skin a field-dressed deer at our processor. It sounds like the skinning is much easier with a hanging deer?
I would never skin a deer on the ground. Not sanitary, even by my Grizzly Adams standards.

5.) Do you actually butcher the meat cuts yourself? Do you keep the backstraps and loin cuts as steaks, and then take the rest of the boneless meat to the processor to make summer sausage and slim jims?
Yes, when I'm done there's a pile of meat and a pile of bones. I cut the backstraps to a length that will fit in a quart ziplock bag. Two or three pieces will fit
depending on the size of the animal they came from. I generally make those into pastrami. I take however much boneless meat that I want for sausage to the locker plant. The rest of it I grind. I've got a size 22 grinder that I drive with an electric motor. You can see the motor stand in the picture I posted. I've got a stuffing horn that attaches to the outlet of the grinder. Muscle goes in, burger comes out and is stuffed into poly sleeves that I close with a small hog ring.

6.) My wife will cook any fish or game that I bring her. However, it must be presented to her as "meat". She wants no part of any blood or guts! Any advice on handy features for when I finally build my work shed? (For deer AND fish guts.) Floor sluice/drain, removeable mats, etc.?
I use rubber stall mats on the floor of my shop, but if a piece of meat hits the floor it goes into the trash. When hunting season is over I drag the mats outside, clean the floor and mats with the power washer, then put the mats back in. When I butcher fish the guts go into a 5 gallon bucket, then get buried in the garden. Once I finish landscaping my pond dam I'm going to pour a foundation for my pump house. Phase two of that project is going to be building a shed onto the east side of the pump house. That will have one wall on the south side, with a countertop running the length of the wall. I picked up a double-basin stainless steel sink at a yard sale awhile back that I'll set into the counter top. I'll run a hose to the sink supply and pipe the drain water onto the ground outside and downhill from the fish cleaning station.