Connect to LAN server remotely?

You need to do what is called port forwarding.

"Open to LAN" with the minecraft client is only intended for local LAN use. It uses a different random port each time you run it, so each time it was run, you would need to figure out what port it was using and port forward through your router to it. You also have very limited control, depending upon initial world settings, either everyone could use cheats, or nobody could, and no way to limit who could connect.

So it would be best to run a regular minecraft server if you want someone outside your LAN to connect to it. Then you could use "server.properties" file to set a fixed port (default 25565 or something else) to make port forwarding easier on your router. You can also set other things, like who can use cheats (ops.txt) enable white-list.txt, banned-players.txt, banned-ips.txt, etc. You need to run it once for it to configure itself and add those files. But after that you can copy a world folder from your minecraft client to the folder containing the server and point server.properties (level-name=) to that world folder name.

It is probably best to have a 64-bit OS and enough RAM if running client and server on same PC. But I have run just the server on a 1 GHz 2 core tablet PC w/2 GB RAM, so you could run the minecraft server on most any old computer more capable than that.

See http://minecraft.gamepedia.com/...p_a_server

You might also want to look into some sort of dynamic DNS if you have a public dynamic IP that could change, to make your home internet connection easier to find by a fixed name. I have used noip.com, but for others web search "dns hosting".

Either you need some remote desktop utility (Windows or Linux), or just run WinSCP (Linux) to access all the files on the server, and run any terminal session (Linux via PuTTY.exe)