Question: How might I implement a local HTTP server using either Java, C#, C or purely Mathematica?

It should be able to respond with Mathematica input to GET and POST requests ideally on W7.

This is related although [doesn't really work.](https://mathematica.stackexchange.com/questions/72551/reading-from-a-socket-stream) If you would like you can read the license [here](https://mathematica.stackexchange.com/questions/27785/implementing-a-100-mathematica-http-server)

Answer: 
Starting in Mathematica 12, there is a built-in function [`SocketListen`](https://reference.wolfram.com/language/ref/SocketListen.html) that can start a web server and respond to HTTP requests.

`SocketListen` is also available in Mathematica 11.2, but only on an experimental basis.

Further reading: [Network Programming Guide](https://reference.wolfram.com/language/guide/NetworkProgramming.html).
Yes (and using pure Mathematica or WolframEngine)

<https://jerryi.github.io/tinyweb-mathematica/>