Coyotes can and do interbreed with dogs, the results are known as coydogs. For the same reasons that interbreeding dogs with wolves is a bad idea, so is this. The offspring are timid and less friendly than dogs, and unpredictable.
Yes, and sometimes it is done on purpose by someone who has domesticated the coyote
I have never seen nor heard of any such records but I would not doubt that somewhere a stray dog has mated with one somewhere in the united states. There are just to many strays to believe that it has not happened.
Although both belong to the canine family, there are no records about a coyote interbreeding with a dog yet a hybrid between the two may exist in a lab whilst performing artificial insemination.(this process was done on lamas and camels and produced a hybrid that wouldn't have normally existed(since both species are too anatomically different to mate on their own).