Do they have package options on what they do with your deer? What options you have for keeping the hide and head? Do you just show up and drop it off? I really don't know. How do you go about it when you're hunting a long ways from home? Can they ship the meat to you? How does that work?
Most lockers will charge a flat fee for basic butchering and packaging. They may ask a few basic questions re. how you want things cut but you can just explain that it is your first time doing this and they will go over it with you. All lockers require a deposit, some require full payment at drop-off. Any processed stuff (sausage, jerky, sticks, etc.) will be an additional charge per-pound. That stuff gets expensive so beware.
All this is done when you show up to drop the deer off. They will check your tag, copy your info and give you a receipt. Unless you are in the South, I would not try showing up with a deer that has not been gutted. Otherwise the drop-off is simple.
As far as antlers go? Most lockers will saw the rack off and set it aside for you for free. A cape for a shoulder mount will usually cost a few bucks, but they are going to do a better job than you will as a first timer. If you are having a mount done, don't be cheap.
Lockers around here will not mess with shipping your meat. You'd need someone to pick it up and UPS it for you. My guess is that outfitters and lockers in areas that get a lot of visitors will be set up for this....at a price.
I'm also wondering what you do with coyotes. What would you do if you're not interested in thier fur? Are there places you can give them to someone that wants them?
Coyote furs have a market value so unloading them should not be an issue, once you find the right people. You may want to start by contacting a taxidermist in the area and see if he or someone he knows will take the furs. If you are killing them in small numbers selling them may become too much hassle and you might be better off just finding someone who will take them off your hands. Wrap them in plastic and freeze them whole until you can hand them off.
As with anything, consult your local regs as well. Some states treat coyotes as a nuisance and have very few regulations regarding them. Other states treat them as a fur bearing animal that has seasons and registration requirements.