1. Did you hear that news last night?
- Yes, I did. That is so heartbreaking.
This is correct. It means that the news affected you negatively, and you were very sad and in anguish.
2. That was the worst way of getting dumped.... My heart is broken.
This is a metaphor. It doesn't mean your heart stopped working. It means disappointment and disillusion from a relationship.
3. My heart is breaking(?) : I think this is wrong, but many native speakers still use this.
I don't really hear this too. Probably the meaning is the same as 2. but it is used in continuous tense. Perhaps something hurtful is unfolding in front of you and it is hurting you.
4. You are breaking my heart!
This sounds more natural than 3. Someone whom you love is inflicting hurt on you.
Thank you :) in advance
Hope this helps.