Hi Tina,
Welcome to BuddyDev.

I may not have understood your requirement completely, but here is what you can do. Please put the following code in bp-custom.php

1. Change @ name to display name for logged in/displayed user

function buddydev_show_fullname_for_displayed_user( $username ) {
	return bp_get_displayed_user_fullname();
}
add_filter( 'bp_get_displayed_user_username', 'buddydev_show_fullname_for_displayed_user' );

function buddydev_show_fullname_for_loggedin_user( $username ) {
	return bp_loggedin_user_fullname();
}
add_filter( 'bp_get_loggedin_user_username', 'buddydev_show_fullname_for_loggedin_user' );

2. Change the label username to User Login

//this is a bad idea, use translation instead
function buddydev_change_login_labels( $translated_text, $text, $domain ) {
	
	if ( $text =='Username' ) {
		$translated_text = 'User Login';
	}
	
	return $translated_text;
}
add_filter( 'gettext', 'buddydev_change_login_labels', 20, 3 );

For your 3rd step, I believe that is not required. Any number of people can have same name and will use it for display name. So, I don’t think that is a good idea.

Hope that helps.
Regards
Brajesh