As I understand it Linux Containers are referred to as light-weight virtualization; virtualization without all the extra OS for every service that is run. Docker utilizes Linux Containers, and Docker has the possibility of running on Snappy Ubuntu Core.
Does the stack mentioned above require the processor it is running on to have VT-X or other virtualization technology? Or is it just a reorganized way of running services on Linux, that doesn't require a processor with virtualization?
No, you do not need any of the hardware virtualization technologies. Those are for full hardware virtualization.
Container, aka. Operating-system-level virtualization, is provided by, in case of Linux, Linux kernel. It's all software thing.