What are the possible states for a docker container?

Are there states other than "running", "paused" and "stopped" or are these the only states available?

No, there are other possible states. The Docker Remote API defines the following states:

• created A container that has been created (e.g. with docker create) but not started
• restarting A container that is in the process of being restarted
• running A currently running container
• paused A container whose processes have been paused
• exited A container that ran and completed ("stopped" in other contexts, although a created container is technically also "stopped")
• dead A container that the daemon tried and failed to stop (usually due to a busy device or resource used by the container) - Added on v1.22