Google-sheets – Copy Google form and linked sheet preserving existing formulas and formatting

If you do not want to create a new "form responses", make a copy of the forms (Not the response sheet). In the copied form, Change the response sheet to the one with all the formatting by going to:Menu>Responses> Change responses destination

This can be done with a script. Enter the following in Script Editor, found under Tools. Change the 'Name' to whatever is the name of the form field that should be used.

function formSheet(e) {
  var sheetName = e.namedValues['Name'][0];
  var sheet = SpreadsheetApp.getActiveSpreadsheet().insertSheet(sheetName);
  sheet.getRange(1, 1, 1, e.values.length).setValues([e.values]);
}

Then set a trigger using "Resources -> current project's triggers" in the menu. It should be "from spreadsheet", "on form submit".

This function also copies the submitted value to the first row of the new sheet.

Note: there cannot be two sheets with the same name, so if some name is submitted more than once, sheet creation will fail.

References: Event object, Sheet class methods.