Hi,
Try looking at /var/log/messages to see if you can spot the problem. At
least it might tell you what happens right before the lockup.
I think the answer may ultimately lie somewhere between Fedora and
Ubuntu on the compatibility question: Fedora, because that's what Qubes
2 is based on; Ubuntu because they have the most extensive database of
certified hardware (by far) of the desktop distros, so I would tend to
look to their HCL for an indication on whether there is hope for running
a /current/ Linux on a particular model of computer. Unfortunately, X200
is not in the Ubuntu HCL even though the X201 is. There's also no
mention of Linux compatibility (good or bad) on the X200's ThinkWiki.org
page.
If I were to investigate further, I would first burn a copy of Fedora 20
x64 KDE or Xfce to disc. Then update the X200 BIOS to the latest version
(if it isn't already), then reset BIOS settings to default, and then I
would disable AMT. Finally, install Fedora to see if the lockup problem
originates there.
You could go even further and configure Fedora with Xen (Qubes load Xen
along with Linux kernel at boot time, and its Xen that provides VM
features)...
http://wiki.xen.org/wiki/Fedora_Host_Installation