Do all flights with the exception of military show up on FlightAware?
No.
From the Flightaware FAQ:

Can FlightAware track VFR flights?
In the US and Europe, this is not officially supported although some VFR aircraft with flight following are available on the position maps but it is largely unreliable and arrival/departure/flight plan data is often not available.

and

Can I track NASA's Space Shuttle or other space ships (UFOs, etc) on FlightAware during launch or descent?
No, but that would be rather amusing; the shuttle or UFOs would have to be on an IFR flight plan to be tracked or have an ADS-B OUT transponder. However, NASA streams live NASA TV on the Internet.

and

Can I track military aircraft? What about presidential movements in Air Force One?
FlightAware does not track military aircraft and presidential movement flights (e.g., Air Force One, Marine One, etc.) are operated by the US military.

etc.
Also compare this screenshot from FlightAware with a screenshot from another free tracking service:


You will notice that several flights are missing on the FlightAware map. The other tracking site, however, is not perfect either; these kind of internet tracking sites mostly rely on ADS-B, and do not receive radar data. Even if they did, we also still have flights which do not even show up on radar. Aircraft surveillance is complicated ...