Which action hook should I use to intercept a form upon submission?

Im not entirely sure there is such action hook for custom forms (someone else might know) but there are definately other ways if form is submitted normally, without ajax.

Let me know if you have any questions.

//This applies if you want it to happen only if form is submited and page refreshed
//It doesn't work if user comes back later - there's no extra field later on

$form_submitted = false;

//If submit is pressed
if( isset( $_POST['submit'] ) ) {

    //There should also be a nonce check here

    $form_submitted = true;      
}

//All your form html here

//Check if form is submitted
if ( $form_submitted ) {

     //Show your extra fields
}



/*If you want it to be a bit more "permanent"

1. update_usermeta() to save the bool that user has submitted the form - logged-in users only
2. Use cookies which might not work properly in browser "stealth mode" and is gone if user deletes browser history with cookies 
3. Use sessions which destroys itselves if user closes the browser

*/