How to properly delete and rename the custom settings/section/control in theme customizer?

You have the value stored in the database in the cta_settings theme mod. If you want to rename your setting, then you’ll need to move the theme mod data you have stored in the database. For example:

if ( get_theme_mod( 'cta_settings' ) ) {
    set_theme_mod( 'phone_settings', get_theme_mod( 'cta_settings' ) );
    remove_theme_mod( 'cta_settings' );
}