WatchMessage works without the use of a third-party server. There is no personal data being sent to me or any other third parties for the app to work. WatchMessage simply connects to Facebook in order to link your account, without any party in between.
To improve the app, anonymous analytical data of the app's usage is being monitored.
There is none of your personal data being stored and read by the app. The only type of information that's accessed by the app itself is preferences such as font size, theme colors etc.
As no data is being stored externally and all data stays on your iPhone and your Watch, simply uninstalling the app will remove the reference between the app and its data. You can also use the "Logout" button to delete the local data.
Yes: Because the app connects directly to Facebook, basically no one but you and the people you open your messages on Facebook to. For information regarding the connection between your iPhone and your Watch, refer to Apple's documentation (it's also really safe from what I can tell).