I have two questions
A .I wonder if the meaning of the next sentence is "1" or "2".
I'm happy being single.
1. I'm happy about being single.
2. I'm happy that I'm single
B. I wonder if "being" is gerund or participle.
I'm happy being single.
It basically means "I am quite content with the fact that I am single. I am not desperate for a boyfriend/girlfriend/partner".
It doesn't mean "I am happy specifically because I am single".
We sometimes use "happy" to mean "I'm OK thanks. I don't need anything/I don't need what you're offering!" For example:
Mother and daughter are eating a meal.
Mum: Do you want anything else?
Daughter: No, I'm fine.
Mum: Really, I have lots more food in the fridge. Are you sure?
Daughter: Yes, really, I'm quite happy with what I have on my plate. Thanks, though.
So here it's closer to "satisfied" rather than "happy - opposite of sad".