Github repo name and local folder name for repo

There is no reason the GitHub repository, your local clone, and any other copy of the code couldn't be named differently. There's no real advantage or disadvantage to it, other than you and your collaborators' ability to keep it straight.

git clone <url> <path> ;# path is a variable; it's the url that's well-defined
git remote add <name> <url> ;# as can be seen in the 'git remote add' command

You can do what you want. The local and the remote repository can have different names.

If you already have a local repository, you can add a remote (named as you want) with:

git remote add origin http://distant/a_remote_repository

If you already have a GitHub repository, you can clone it in the local folder you want with:

git clone http://distant/a_remote_repository a_different_local_folder