Need to plan ahead what to do with Coyote.

Most taxidermist is able to cure hides I would find a local one and explain your concerns with him or her, they can tell you what they will want when the hide comes to them.
Please note that under many state laws the taking of an furbearing animal due to pest issues and due to collection of fur fall under two separate laws and regulations.
In my state for example If I shoot a bobcat as a pest, I can keep the fur for myself but cannot attempt to sell it or ship it outside the state. Also If I shoot a bobcat for fur it is not or may not be legal to do so with an air rifle so make sure to check your local laws before you start planning to far ahead.
As to skinning or casing it depends on what you want to do, If you are making wall trophies for yourself then I would suggest you go ahead and dig in its the best way to learn Wall trophies are flat like a rug and fairly easy to do.
Go to Walmart and in the hardware section you will find those little red breakoff razor knives for like a .25 cents a piece grab a handful they are the perfect choice for skinning. 
Again talk to the taxidermist and he will show you, if you choose to have someone else do it then yes you need to at minimum field dress (gut it) and put it on ice asap to keep the fur from slipping

If you field dress (gut it) you will be cutting along the belly. This will make it more difficult for the person fleshing it. Coyotes are a little tricky to skin, but fleshing is much easier than a raccoon. If you don't want to do it yourself, I would suggest putting it in a trash bag and straight into the freezer. Personally, this has worked out for me. If you do it yourself make sure you skin it warm. 

5 gallons water
one lb. alum (aluminum sulfate for pools)
1 1/2 lbs.salt
Neat's foot oil

Dissolve alum and salt in water in at least a 10gallon capacity plastic or rubber container. Remove all meat and fat from hide and place it in the solution. Stir and move the hide around twice a day for five days. Make sure all parts of the hide are thoroughly wet. After five days,remove the hide and rinse it thoroughly with a hose. Apply Neat's foot oil to the skin side only not to the hair. As the hide begins to dry work it stretching and kneading over the rounded edge of a board to make it pliable. Re-apply Neat's foot oil while working the skin side back and forth over the board(saw horse works well). Finally, when the hide is pliable and fairly dry, use coarse sandpaper to soften the skin surface and clean it.

Foxes have a very pungent musky odor that gets a little nauseous to me. Deer are a charm. Beaver just stink. Coyotes are just dirty dogs and a gator makes me puke. Hahaha! Fox skin hat using the head is cool. Especially the ears standing up. You can cut up the hide to make knife sheaths etc using strips of the hide to sew, or leather laces.

I've done every critter imaginable, Haha! It is work. Some prefer to send it off and pay, and a reputable company will send it back with beautiful results. That is expensive, but sometimes worth it. Snake skins do well with borax, salt and glycerin, but thats another recipe for another time! ; ) Enjoy, live a little- Plinker!