They can do /join "server-name" (Anyone can start any server). If it gives them (or you) an error message, relog, wait a lil bit, and try again. Repeat this process and you will eventually be able to join the server.
Well I'd say it depends on the amount of popularity the event gains. If less players play, there shouldnt be some grand prize. If you were able to partner with other MH servers, they could offer VIP or smth to the winner of a larger MH Monday.
So this works with all commands that would use @, but I'll use tp as an example.
You need to do /minecraft:tp @p
Adding the "minecraft:" to the beginning tells the server not to adjust the command to the plugin. Hopes this makes sense.
There's not an option to "disable diamonds" unless you do this manually with world edit (//replacenear 10000000 diamond_ore air). As Gaminger said, you probably just need to mine deeper. Usual Y-coordinate for diamonds is 17.
Timing out is a network issue (Either on your end of the server end). There's not really a way to stop this from happening because it happens to everyone at some point.