1)"I am thinking of visiting New York next year."You are considering going there, but you haven't decided for sure yet.
2)"I am thinking his name was bill"You think his name was Bill, but you are not sure.
3)"I am wishing I had stayed home."You have gone somewhere, and you regret going there. You wish you hadn't left home.
4) I am hoping to take my exam soon. It is your wish to take your exam soon (without much delay).
5)I was hoping (that) there would be a few tickets left for the jazz concert.You are expressing disappointment that there are no tickets left for the concert.