I don't know how much this will help but my son has been having similar problems.
Using a monitor connected to either VGA or DVI, the PC will boot, beep and then sit at a black screen with a mouse cursor only.
One thing I discovered, is if you connect an HDMI monitor (leave the VGA or DVI monitor connected, very important) then when you turn on the PC, it will now boot normally. Both monitors will work.
Now here is the weird part, if you un-plug the HDMI, the VGA or DVI monitor will go dead.
Plug the HDMI back in, both monitors come back.
We called Alienware support and they say we need to uninstall the NVidia drivers and install the latest ones from their web site.
If you click the 'System Scan' option, it will report that you don't have any NVidia hardware installed.
If you manually enter the NVidia information, it will give you a download. When you try to install the drivers, it tells you that your NVidia hardware is incompatible with the drivers.
We are running Windows 10 Pro x64 and tried those drivers, along with Win 8.1 drivers just for fun. They don't work.
My son is happy for now because Windows 10 has a NVidia driver that seems to work and he has a TV which accepts HDMI so he just plugs that in when he plays, still using the VGA monitor.
We will be calling Alienware support again. :-(
Hope this shines a little light on the issue, although it doesn't fix it.
Just a note: We also re-installed Windows 8.1 and Windows 7, both of them have the same issue, so its not the OS. Also, Alienware admitted that it's a design flaw in the hardware that causes this and the allusive drivers will supposedly fix it.