Garry's Mod server status checker
Garry's Mod answers on the standard Source query protocol, so a failed check almost always means a firewall rather than a game problem.
Which ports a Garry's Mod server needs
The query runs over UDP on the same port players connect through, which makes Garry's Mod simpler than most games here — one port, one protocol. A rule that covers TCP only will pass a casual test and still fail every query.
If it still shows offline
If the port is open and the query fails, look at the addon list. A server that is still mounting a large collection at startup does not answer queries yet, and a Lua error during startup can leave the process running without the game ever becoming queryable.