There are a lot of depends answers to your questions.
If you are somewhere very cold and freeze them whole, you can take your time skinning them. If you skin them and roll them right so you don't get freezer burn they can be stored like that a long time.
If you skin, flesh, clean, Form, sew and "put up" they can store for a long time in the right conditions.
For how to skin, Because buyers want it a certain way.
http://www.youtube.com/watch?v=p9HujqIeGl0&list=UUPuxmoN5GsVG84BhIO3ScSA
Fleshing
http://www.youtube.com/watch?v=E56TbpYR8lg&list=UUPuxmoN5GsVG84BhIO3ScSA
Basically just watch his video's on what to do from start to finish. Other people will have other ways of doing it. But, he puts up over 200 coyotes and sends them in to sell. You will find what works for you, but it's a good starting point.
If you can't find a buyer local, then you may want to look into sending them to NAFA and creating an account. Some people prefer this and if you batch send the pelts in, saving on shipping, then you could save some money. I am a toss up on this one.
He has a video of watching the auction. It's kind of a lot to take in.
Here is my Junior i will defer to others that have more experience on it understanding.
You create the account, and then when you send in your pelts, they store them for the scheduled auctions that are coming up, and then they grade them. Once they are graded they are put into the lot's and then the auction you can watch is of each lot and the sale price per pelt.
Once it's all said and done, you get your check (minus fee's) for each coyote you sent in that was in each lot.
The up in the air, is the shipping and the handling fee's. Your local buyer may pay you $75 for a prime Montana Coyote. That is a check right there that day. They send it to one of the auction places. Where you could set up the account yourself and send it. The auction goes off and instead of getting $75 the buyer gets $100 or $65 it's an auction and gamble.
Just something to decide.
Or, you could send them in to be tanned and make things with the hides and sell them to the local shops that get tourist or on line.
Either way, good for you for doing something with the pelts. I personally feel that it's respectful to use the pelts for something. As long as you can at least break even for your time. That all depends on the person and what they feel their time is worth.