InDesign Script new layer naming by page number

Ok, here is a solution that should solve your issue or at the very least send you on your way. As I described in the comments below the post, this sets up a layer that is dedicated for exporting, duplicates one item after the other to this layer, exports and removes the element again:

// set the preferences for the PNG export
var pngPrefs = app.pngExportPreferences;

pngPrefs.pngExportRange = PNGExportRangeEnum.EXPORT_RANGE;
pngPrefs.pngQuality = PNGQualityEnum.MAXIMUM;
pngPrefs.exportResolution = 300;
// etc.; for all possible settings see
// https://www.indesignjs.de/extendscriptAPI/indesign8/#PNGExportPreference.html

var doc = app.activeDocument;

// loop over all layers and make them non-printable

for (var i = 0; i < doc.layers.length; i++) {
  doc.layers[i].printable = false;
}


// set up layers
var exportLayer = doc.layers.add();
var sourceLayer = doc.layers.item("Answers");


// loop over each page item, duplicate it to the export layer and then export it

for (var i = 0; i < sourceLayer.pageItems.length; i++) {
  var pageItem = sourceLayer.pageItems[i];

  // duplicate the page item
  var dupItem = pageItem.duplicate(exportLayer);

  // set up export file
  var expFile = File('~/Desktop/output/page' +
      pageItem.parentPage.name + '-' + i + '.png');

  // set which page to export
  pngPrefs.pageString = pageItem.parentPage.name;

  // export the document
  doc.exportFile(ExportFormat.PNG_FORMAT, expFile);

  // remove the duplicate item
  dupItem.remove();
}

Edit: To avoid the error with page items on the pastboard, you can quickly check, if the item has a valid parent page. You could add this snippet below the var pageItem ... line:

if(!pageItem.parentPage) continue;

This basically tells the script to skip the current page item and continue with the next.

How about just adding as many new layers as the maximum number of items on a page? That way no page would have empty layers.

// main function
function distributeItemsToLayers() {
    // active document
    var doc = app.activeDocument;
    // source layer
    var sourceLayer = doc.layers.item("source layer name");
    // array for new layers
    var newLayers = [];
    // loop through pages
    for (var i = 0; i < doc.pages.length; i++) {
        // page
        var page = doc.pages[i];
        // array for items on page in the source layer
        var items = [];
        // loop through items on page
        for (var j = 0; j < page.pageItems.length; j++) {
            // item on page
            var item = page.pageItems[j];
            // check if item is on source layer
            if (item.itemLayer === sourceLayer) {
                // add item to array of items on source layer
                items.push(item);
            }
        }
        // loop through items on source layer
        for (var j = 0; j < items.length; j++) {
            // item on source layer
            var item = items[j];
            // check if a new layer exists for this number
            if (newLayers.length < j + 1) {
                // add new layer for this number
                newLayers.push(doc.layers.add());
            }
            // set item layer to corresponding new layer
            item.itemLayer = newLayers[j]; 
        }        
    }
}
// run main function as one undoable action
app.doScript(distributeItemsToLayers, ScriptLanguage.javascript, undefined, UndoModes.ENTIRE_SCRIPT, 'Distribute Items To Layers');