Yes, in the U.S. they are pronounced identically, as far as I know. I'm no phonologist but I don't know of any regional accents in which they are different. When I was a kid and I heard the word "mourning" spoken aloud I thought it WAS "morning" and found it confusing.

Yes.

In standard British RP (Received Pronunciation)/ BBC English, yes, they do have the same pronunciation. This isn't necessarily the case in all pronunciations, however. I think they would be different in Scottish and Irish English, and may well be different in some American accents.