When a woman gave birth to a baby, she can say, "I had a baby." Can her husband use this sentence and does it sound all right? If someone is talking about her son, can she say, "My son had a baby yesterday"?
The man can say, "My wife just had a baby," or, "We just had a baby." (In the latter case, it's pretty evident that the wife went through the tough part, although the husband may have been a labor coach in the delivery room.) But, at least in the U.S., most men wouldn't say, "I had a baby" when referring to the childbirth itself.
I've heard both of those expressions used. As I mentioned, men won't usually say "I had a baby" when referring to the delivery, although they might say, "I have one child" when referring to their status as a parent. A father might also say something like, "I had two boys and my wife had two girls" when explaining how his blended family got together. In that case, once again, the verb had refers to the man's parental status, not his experience in the labor and delivery room.
One situation where you might hear a man saying, "I had a baby" is when the couple had a child who passed away in infancy (as in, "I had a baby, but she died when she was six months old"). But that's a different context.