Yes, you can use it to answer 'how are you?' but it's not a synonym for 'really happy.' It means that you feel really good and well in yourself and/or that your life is going well at the moment. It's not really a comment on your emotional state. So synonyms for 'phenomenal' would be things like 'fantastic' or brilliant.' As an aside, when people ask the question 'how are you?' they're not normally asking about your emotional state, whether you're happy or sad etc. Instead it's more courtesy to enquire about how you are doing generally. Synonyms for 'really happy' would be things like delighted, ecstatic or even 'over the moon.'

You could, but it is very Hollywood, very drama-Queen and a little fake. It is like a dress, and if you choose to use it, you need the right personality, looks and lifestyle to match. Otherwise, "I sm very well, thank you" would be a much better response.

Yes, when someone asks how are you, you can say I'm great or I'm phenomenal.

You can use "phenomenal", you could also use "fantastic" or "brilliant". :)