1. 22-250 or 243 is what i used.
2. sit and wait and if you're lucky you'll see one...or 4. you could also call them, but i have no experience with calling.
3. depends when they are out looking for food. oddly the 5 coyotes that were hunted by me and my girlfriend's dad were shot at all different times. 10am - 9pm was the range.
4. skin it if you want to sell the fur. eastern coyote fur isnt worth as much as western, but you might get more if you wait for their winter coat to come in. the carcus went to the bone pile for me.
5. not sure about michigan. in ohio, the law is no limit, all year round. you need a general hunting permit though.