SharpBox is using the official API interface for 3rd party application (which is not the same the DropBox client is using). The DNS names for the communication are
api.dropbox.com --> Only Metadata --> Directory and File Browsing
api-content.dropbox.com --> Mass Data --> Downloading or Uploading files
Since 1.0.3 SharpBox is using SSL only (if you not disabled this) so port 443 should be enough.
As I know DropBox is using Amazon as a service provider for the infratructure so could be that the API servers are running in Amazon.
I hope this helps