Not quite. 'He is bound to be happy' is talking about future happiness. For example, 'when he sees his grades he is bound to be happy'. 'He is surely happy' talks about present happiness. For example, 'I heard he received good grades, he is surely happy'. I would say that I haven't heard many people use 'he is surely happy', in my experience people are more likely to say 'he must be happy' or 'he's gotta be happy with that' or something similar - it just indicates that you are guessing the person is happy. Whereas 'he is bound to happy happy' means you are guessing he will be happy at some future time.

He is bound to be happy. = He is certainly going to be happy. Another variation would be: He is destined to be happy. Hope that helps.

More like, "He will surely be happy." It's a prediction.