Google-sheets – Data Validation for Sheet1 from Sheet2 list

=COUNTIF($A:$A,"="&A1)  < 2

If you put this in as a Custom Formula for the Data Validation rule for Column A, Column A will reject all duplicates.

I have expanded the functionality of the script from the linked question accordingly. First it checks whether the sheet with given name already exists. If it doesn't, the sheet is created and the headers of the form responses sheet are copied there. Otherwise, the existing sheet it used. Either way, the newly submitted data is then copied to the sheet.

function copyData(e) {
  var sheetName = e.namedValues['Name'][0];
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var sheet = ss.getSheetByName(sheetName);
  if (!sheet) {
    sheet = ss.insertSheet(sheetName);
    var formSheet = e.range.getSheet();
    var headers = formSheet.getRange(1, 1, 1, formSheet.getLastColumn());
    headers.copyTo(sheet.getRange(1, 1));
  }
  sheet.appendRow(e.values);
}

The function must be equipped with a trigger using "Resources -> current project's triggers" in the Script Editor menu: the trigger options are "from spreadsheet", "on form submit".