Do you save the user's key in server to decrypt? If that's the case, how are the notes truly 'private'?

No, not at all. All encryption/decryption happens client-side. That is why it's zero-knowledge.

The key stays with the user only. We have no knowledge of it.

How does this handle conflicting offline writes?

For most cases, the most recently edited copy is kept. However, if there's a conflict between 2 notes then a nice Git Conflict like UI is shown with both notes side by side and their differences highlighted.