I would think about it a little more directly. First, J is an ideal because any polynomial multiplied by an element of J will give you an element of J. It is principal because it is generated by x.
For the second part, I can't seem to give a rigourous explanation that I am happy with. Try to write a few of the cosets.
They are all different and there is an infinite number of them because you can just keep raising x to a higher power. I hope I didn't confuse you with that. I'll think about it some more and try to come up with a better explanation.