Google-forms – Can one Google Form store a constant to be called and resubmitted in another

You can run a script that will catch the submitted responses, then add a row to the public spreadsheet with the appropriate data. If you need any help writing this, let me know.

Edit

Something like this should do it:

/**
 * This will 
 * set a trigger for this as described here: 
 * https://developers.google.com/apps-script/understanding_triggers#ActionTriggers
 * @param {Object} e The event parameter for form submission to a spreadsheet;
 *     see https://developers.google.com/apps-script/understanding_events
 */

function onSubmit(e){
  properties = {publicSheetId:'PUT YOUR PUBLIC SHEET HERE',
                publicFields:['Question 1', 'Question 2?']};
  var form = FormApp.getActiveForm();
  var responses = form.getResponses();
  Logger.log(responses);
  var lastResponse = responses[responses.length -1];
  var publicSheet = SpreadsheetApp.openById(properties.publicSheetId);
  //form = FormApp.getActiveForm();
  //e = FormApp.getActiveForm().getResponse()
  Logger.log(lastResponse.getId());
  var itemResponses = lastResponse.getItemResponses()
  var publicItemResponses = []
  for (var i = 0; i < properties.publicFields.length; i++){
    var publicField = properties.publicFields[i]
    for (var j = 0; j < itemResponses.length; j++) {
      var itemResponse = itemResponses[j];
      Logger.log(itemResponse.getItem().getTitle());
      Logger.log(publicField);
      if(itemResponse.getItem().getTitle() == publicField){
        publicItemResponses.push(itemResponse.getResponse());
        break;
      }
    }
  }
  publicSheet.getSheets()[0].appendRow(publicItemResponses);
}

Working on the solution:

...a check that will make sure the email address that they provide actually exists in the organization...

We were facing a similar situation. Being an educator we have youngsters that quite often misspell their email accounts when filling out their test forms. As a result we would end up sending their grading to the wrong or even a non-existing email.
We are not script savvy so we came up with a workaround.

Instead of having them fill out their emails, we provide those emails as a dropdown list (taken from a class roster) and all they have to do is choose their own.
Assuming the organization is not a couple hundred people (in which case you most definitely need a script), this could very well work in your case as well. If you were to change the droplist to a checkbox, you could even have someone send a greeting card to more than just one member.

You may argue the above answer as the "right" choice. The reason being it may cause a security breach. In addition to having personal data publicly exposed. In that case you could sent the form by email or even have it password protected.

A scripting solution that could, after submitting the form, compare the answer of a specific question to a given list of emails and take an appropriate action (e.g. sent an email to the form creator for a not-listed email) would indeed be very welcomed by us as well.