How to get a free, editable domain?

Assuming the server software is free or you own a licence already (and a client), you could set up a web-server on one of your own sufficiently powerful computers.

If you want to run a minecraft server at home, what limits how many people can connect without too much lag is your "upload" speed which is usually significantly slower than your download speed. If you have DSL you may not be able to have more than a few people connect, but I have known people with fast cable or Verizon FIOS run minecraft and/or certain mods like Tekkit servers.

If like most people you have a dynamic public IP, there are dynamic DNS services that can resolve a free fixed subdomain to your dynamic IP address. You either need a router capable of supporting that or run a service on a computer that accesses the dynamic DNS service periodically to grab your connecting public IP to keep the DNS up to date. I have used noip.com, but for others web search "dns hosting".

One of the first things you would want to do is set up a server at home and make sure that you can actually figure out how to run it and be able to connect to its LAN IP locally with a minecraft client on another computer (to make sure that a firewall is not blocking that on the server. Note that "Open to LAN" on a minecraft client will not allow people to connect to it from outside of your LAN like a regular server will. It is best to leave server-ip= blank, so it can bind to any IP on the server, instead of specifying a wrong IP that may not work.

Once the server is working locally, you need to configure port forwarding on your gateway or router for port 25565 (or whatever you set for server-port= ) to LAN IP of your server. See if http://www.canyouseeme.org/ shows that port open (with the server running).

A server somewhere else out there on the Internet with faster connection speed for more players is unlikely to be free. But if you are going to go that route, make sure that you get some practice running a server at home first to become familiar with the basics of running a server.