I've had the server up and running for a few days now, and zero (not a single one) players have joined.  I know there are lots of servers, but this seems HIGHLY unlikely.  How do I make sure that my server is showing up in the game's server list properly?

The in game Server Browser (For Rust and other games) are designed to query the steam service engine, and retrieve a set of details based upon a prebuilt list of factors. For Rust, the query is primarily if the server is a community or modded server, as well as the server version, and the ping response time between a steam data point and the servers in question. 

This is NOT the game client's ping to steam, or the server's actual ping to steam, or a connection route to your player base, but it's a traceroute response from the Rust central server, to Steam, to the server, and then round trip back.

While it's retreiving that data point, the Rust browser builds that the list of servers that respond within a preset of metrics, and filters out things that do not fall within FacePunch Studio's parameters.

This list, "times out" after a maximum of around a total 5000 replies, before being sorted. As of last count, in the United States alone there are an "insane" amount of servers, with an estimation of there being an average of one active server for every 3 active players. BattleMetrics (a much more trusted system for retrieving servers to join) times out just after 42000 responses.

As the BattleMetrics system houses at least 14+ times the number of servers than what the actual game can find, and FacePunch Studios has previously admitted that total player count that has been on a server recently does factor into the weight of the results, the in game browser is a HORRIBLe way to try and get your players to join.