Terminal vs. Service?

I'm a longtime MS admin and developer, but totally new to Linux, so the concepts are not unfamiliar, but just how to do this and that are stumping me sometimes. My friend Google has been a great help.

So far I've:
Installed Ubuntu 14.04 on a Dell PowerEdge T610
Installed and run two Minecraft Servers.
Installed xrdp and xfce4 to remote desktop into the server.
Installed and fooled around with Screen.

My questions are around the context of a terminal session.
If use the desktop to launch terminal then launch a persistent process like Minecraft server, does it stick around if I log out? If I log back in, how can I re-attach to that instance to see what's going on, or manipulate the program (for example, add a user to Minecraft).
I think there is a concept similar to a Windows Service in Linux, and if so, should I (can I) launch something like Minecraft as a service?

If I use Putty to SSH to the server, can I then launch a process like Minecraft server then log out, then recoect later to admin or manipulate the program?

Am I thinking about this all the wrong way?

I might be over complicating things here, but I'm also just trying to lea how things work. I spent some time making the remote desktop work, but I realized I really don't even care if it has a "desktop". I just remotely log in, launch a few Minecraft servers, log out, then log in from time to time to see that they're ruing OK.

BTW, none of this is high security, I'm local on my own lan and just trying to run Minecraft as way to lea.

Any thoughts would be appreciated.
Ender