Yes, that's exactly it. It could be a cheap VPS, too, though, like through linode or digitalocean. TNet's server doesn't require a lot of resources, so the cheapest plan should work.
No, it's just the standalone TNet server. Nothing tied to Unity.
The standalone server should be in a zip file within the TNet asset. The zip file is called "TNetServer.zip". It's precompiled so you should be able to just run it.
I think the setup ArenMook is describing is as follows:
You have a single standalone server running. Your players connect to that. Then you split this single server into multiple channels. Each channel is its own separate instance (or "match" in your case) of the game. In this way, each channel kind of becomes its own sub-server. So, in your case, I imagine joining a match currently consists of three steps: TNServerInstance.Start for the host, then TNManager.Connect followed by TNManager.JoinChannel for everyone. With this setup it'd just be TNManager.Connect on startup (only once!) and TNManager.JoinChannel for each match.
Some batch/bash files to startup the server (I don't know what I'm doing here, but it works for me):
Windows (save as RunServer.bat):
@echo off
start "TNet Test Server" "TNServer.exe" -name "TNet_TestSrv" -tcp 5127 -udp 5128 -tcpLobby "127.0.0.1" "5129"
Linux (save as RunServer.sh):
#!/bin/bash
echo "Starting Server"
mono "TNServer.exe" -name "TNet_TestSrv" -tcp 5127 -udp 5128 -tcpLobby "127.0.0.1" "5129"
echo "Server has exited"
For linux you'll need to install mono. I think it's just sudo apt-get install mono-complete. Dunno the yum equivalent, I think you can find instructions on mono's website for a bunch of different package managers.