Google-sheets – How to record the date/time when a new row is added

You could add a trigger that runs when new rows are added to the spreadsheet:

function onModification(event) {
  var sheet = event.source.getActiveSheet();
  if (event.changeType == "EDIT") { // A cell is edited. Other values include "INSERT_ROW" (when a new row is inserted)
    fillEmptyRows(sheet);
  }
}

function fillEmptyRows(sheet) {
    var range = sheet.getRange(1, 1, sheet.getMaxRows(), 1);
    var firstColumnValues = range.getValues();
    for (var i = 0; i < firstColumnValues.length; i++) {
      if (firstColumnValues[i][0] == "") { // Column 0 is A
         firstColumnValues[i][0] = new Date();
      }
    }
    range.setValues(firstColumnValues);
}

You need to register this trigger manually: Click Resources -> Triggers -> Add new trigger, select the onModification method and select on modification from the right-most dropdown menu.

When a cell is edited, the script will fill all blank cells in column A with the current timestamp.

I have set up an example spreadsheet, feel free to play with it.

You just need to wrap an extra conditional around the "stamping" code. Check the row value and only execute the stamping part if the row is NOT 1.

function onEdit() {
  var s = SpreadsheetApp.getActiveSheet();
  var r = s.getActiveCell();
  if( r.getColumn() != 18 ) { //checks the column
    var row = r.getRow();
    if (row != 1) {
      var time = new Date();
      time = Utilities.formatDate(time, "GMT-08:00", "MM/DD/yy, hh:mm:ss");
      SpreadsheetApp.getActiveSheet().getRange('B' + row.toString()).setValue(time);
    };
  };
 };