Not acceptable and unacceptable are the same. Not happy and unhappy often mean the same thing, but are not always interchangeable: "I am not happy" can signify dissatisfaction ('this bread is stale, I am not happy"), i.e. a lack of happiness which could take a number of forms, including anger, impatience, disappointment etc., whereas to be unhappy means to be sad.

I think that "unhappy" and "unacceptable" are more frequently used before nouns. For example, I would say: That is unacceptable behavior. She is an unhappy person. With regard to the examples you gave, they essentially mean the same thing, but I think there may be some nuances. Like others said, "unhappy" sounds like a higher register of speech than "not happy." Also they can be used differently. For example: I am not very happy. I am very unhappy. I am an unhappy person. That is not acceptable. That is unacceptable. That is totally unacceptable! The word one chooses may depend upon the sentence construct, register of speech and other nuances. These things are usually learned by becoming intimate with the language.

They have the same meaning. As a native English speaker, I think saying unhappy, unacceptable, unfathomable, unthinkable, etc.. shows a higher level of English. But remember, not every word can have the prefix "un" attached to it and then it becomes a word. But many can.

The first choices are often used in American language rather than the second ones.

No difference.