Minecraft Java Edition – Can Server Users Download Mods Directly?

I assume you are talking about the Yogscast Complete pack on the AT Launcher. In the GUI, if you use "Create Server", it enables you to open a dedicated server. There is no dedicated download, you just enable and start it through the ATLauncher GUI.

There is no possible way to distinguish MineChat from a standard client. It's the same with many hacked clients, cracked clients, or similar. From the sound of it, neither of those are a problem, though. A thing that you could do is, if you would be willing to run a Bukkit or similar plugin-supporting server, is have a plugin that isolates players from everything until they move. This would deter anybody with MineChat. I could try and show you how to make it with command blocks, but I doubt you want me on your server, as I am a stranger. I hope that I helped in some way. Have good luck.

EDIT: I may have worded my answer in a confusing way; I'm sorry for that. There would be a way to prevent people with MineChat from doing anything on the server, but there is no way to prevent somebody with a cracked client from doing anything. If you wish me to, I could set up an anti-MineChat thing with command blocks, and I could send the MCEdit schematic for you to put it in your world. That's the best that I can do without just building it on your server.