Are LXC, and Docker containers suitable for VPS or for use inside running virtual machine? Do containers have a significant overhead for the running system on VPS or VM?
How secure are applications running in containers compared to services running directly on the host system?
A container is an isolated process on the host kernel, the overhead is minimal, especially compared to a vm. It depends on the capabilities of the vps offering if Docker can be used – ask the provider about it!
Please google for the difference between vms and containers. You should find plenty of blog posts about it.