How can I get VirtualBox to run at 1366x768?
While the VM is running, you can also tell the guest OS to change the resolution by using:
VBoxManage controlvm "Name of VM" setvideomodehint 1366 768 32
The 32 at the end denotes the display color depth.
After burning an entire weekend on this, I finally found the culprit: VirtualBox's 3D acceleration.
If you go into your virtual machine's Settings > Display, and uncheck "Enable 3D Acceleration", then the guest will successfully run at 1366x768.
I solved the same problem on Fedora 22 KDE by using following command:
$ VBoxManage setextradata global GUI/MaxGuestResolution any
$ VBoxManage setextradata "Win 10" "CustomVideoMode1" "1366x786x32"
Note that your virtualbox window should be closed.