you do not have to apply to own or run a minecraft server. if your asking about applying to be a member on a specific server the people on said server (if it has graylist) can tell you. if you found on on a forum or such ask on said topic.
See the related link for the Minecraft forum.
you cannot set up permissions in a normal minecraft server. you must have a bukkit server to use permissions. check the bukkit forum for more info.
It depends on what server you're playing on and how long they choose to ban their members for advertising a server. Each Minecraft server bans differently, because one server might ban a member who advertised for a week, and another might ban them for two weeks. Check the forum or website of the server you go on and look for their banning lengths, or ask on the server's forum.
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The whitelist for a server is a list of usernames allowed access to play on that server. You can't just walk through it. Apply for a whitelist on that servers forum, site or thread
A dedicated Minecraft server is a server where you can play original Minecraft. Usually without plugins or mods.
you need to download the server pack to go on a server in minecraft classic
A minecraft server adress is the location on the internet in which the Minecraft server is stored on. Minecraft server addresses are often a set of four numbers, such as 000.000.000.000(:25565). If the owner of a large Minecraft server has a website, it is not uncommon for the address to be something like this: "hub.worldofkeralis.net".
Minecraft 1.2.3 is a game version, not a server.
The Mineplex MineCraft Server is owned by Bluetomb.
You drag it into the plugins folder. If you want plugins in your minecraft server use bukkit, not the vanilla minecraft server.