You asked a multi-part question, so the answer to your question is yes and no. Yes you can run MC on a headless server without manually logging in, but MC needs an active X session in order to run so there needs to be some kind of user session running (but that can be automated).
By far the easiest way to automate it is to run a cron script either @reboot or once a minute that starts a VNC server if one isn't running (which generates the necessary virtual X session) and then starts MC if it isn't running. On debian-based distros (which typically ship tightvnc rather than tigervnc) it would look like this:
#!/bin/bash
export USER=Username
ps -e | grep tightvnc || vncserver :1 -geometry 1920x1080
export DISPLAY=':1'
ps -e | grep mediacenter || mediacenter21 /mediaserver
If Ubuntu 14.04 uses a different VNC server, substitute the appropriate one; if you're already running a VNC server for remote access you can skip that part. There are other ways to do it that are more complex that have a slightly smaller memory footprint, or that allow the use of RDP or other remote protocols, but this is the method I recommend because it's very simple.
I use a slight modification of this method on my own headless server and it works nicely.
If you need more complete instructions look here:
https://yabb.jriver.com/interact/index.php?topic=99370.msg687832#msg687832 The instructions are geared towards a Raspberry Pi running Debian, but they should work with minor modifications.
EDIT: One additional thing to keep in mind: there isn't perfect feature parity between the Windows and Linux versions, so make sure that all the features you use are supported on the Linux side before migrating your server. The main one to be aware of in a server context is the lack of TV support, but there are others.
