You have correctly created new columns in table messages. But i did not get your purpose to extend the message table for this can break items many places.
You have extended the fields in Tabe Database. You have included the HTML in messages.php file as well. But the form from messages.php takes data to messageprocess.php in includes/ folder , where a function execute the message data. And create a row for table message $message = $message_obj->new_message($message_to, $subject, $message)
So you are not sending data from your html to your database to save. Please do it correctly and if you are not sue how you can do this i recommend you to hire someone who knows php.
I also recommend you to do not modify the table and files of messages. Just extend the table of users for what fields it does not already have and use existing fields as well. And extend user registration form to your required fields which will open a user in without login mode. To register message cannot be posted without login.
And when a user register admin will be notified, and you can adjust from settings what type of user will be registered as admin, as subscriber or what. You can try to extend the table and files of user registration and then you can post message here for me to get a suggestion. Please note customization is not part of support but i will try to guide you thanks