- 나는: I (subject)
- 재미있는: interesting
- 게: thing (object)
- 있어: have
게 is a contraction of 것이.
True, but particles are often dropped, so it's an easy mistake to make.
It's correct, technically. It would be hard to think of all possible translations for this sentence, though
Because there's actually no "one" in the sentence, it's 게, which means thing/something. "I have an interesting one" is incorrect, as that would be something like "나는 재미있는 한 개 있어." (I could be mistaken, but I hope that was helpful)
In English one has a much wider range of meanings than the numeral 1 tends to have in languages: for instance, it can be an impersonal pronoun, as in ‘Everybody was given a book; most found them boring, but I have an interesting one’. I don't think 한 개 would be used in this context, and I'm wondering if 게 might. (I think Japanese would use の, which is very similar.)