Does a docker container have its own TCP/IP stack?
The network stack, as in "the code", is definitely not in the container, it's in the kernel of which there's only one shared by the host and all containers (you already knew this). What each container has is its own separate network namespace, which means it has its own network interfaces and routing tables.
Here's a brief article introducing the notion with some examples: http://blog.scottlowe.org/2013/09/04/introducing-linux-network-namespaces/and I found this article helpful too:http://containerops.org/2013/11/19/lxc-networking/
I hope this gives you enough pointers to dig deeper.