Stress will often indicate the subject of interest. For example:
What does he do? — to indicate that the action itself is important.
What does he do? — often used to indicate that until now you thought “he” did one thing, and it turns out he does something else and you don't know what.
What does he do? — to indicate that you're interested in this particular person's action (perhaps we are talking about a situation in which you usually do something, but you see that “he” gets a better result out of this type of situation, so you're curious about what “he” specifically does).
As a commonly occurring variation: “What are you doing about it?” usually means “It doesn't matter what can be done, should be done, or other people do, it's important what you, in particular, do or want to do with it.”
What does he do? — the common way to ask questions, which doesn’t imply any particularly important detail.
The missing “h” you mention is an accent, a habit of pronouncing sounds. English accent varies a great deal across the various English-speaking territories. Inside the U.S., you would be able to tell a person from Louisiana apart from a person from Boston by just hearing them speak. Omission of “h” at the beginning of a word is very common.