How to create a remote server on LAN instead of using GitHub?

You can set up a remote repository and make it accessible through any of the protocols it supports. The AAA would be handled by the transport.

Yes, you do. Actually you need a SSH-service and git would perfectly work over SSH. Since you're on Windows, see Setup a Git server with msysgit on Windows

There are several ways to do this

1. Host internal repositories like Gitlab or Stash. This will be similar to services like BitBucket or GitHub
2. If you want to have a simple service with SSH authentication - user3159253 has already answered that
3. A very bare bones way is
   1. server: Create a bare repo: mkdir -p RepoName && git init RepoName --bare
   2. server: Start the git daemon: git daemon --base-path=$PWD/RepoName
   3. client:
      Add your remote: git remote add origin git://server.url.or.ip/RepoName
      or just clone it: git clone git://server.url.or.ip/RepoName