The networking features introduced in Docker are one "standard"; CNI is another "standard" [1].

Weave Net as a product ships with support for the former; the blog post describes work towards support for the latter.

No they do not interoperate; there is a detailed argument at http://blog.kubernetes.io/2016/01/why-Kubernetes-doesnt-use-...

[1] Scare-quotes because all of these things are only a few months old.