Is there any way we can have new user registration on Organization Login Page

This would not be possible with the out of the box authenticator.  You would have to develop a custom authenticator,

Yes you can  do this.  You can either create a basic HTML form page that calls the required TWX REST APIs to create a a new user.  You would use an appKey with the necessary permissions to run the user create services and put the user in any required orgs/groups.  You could also create a TWX mashup that you access via the same appKey ( in the URL) and just bind in the services you need to create users and assign orgs/groups.