To validate Swiss Pass on border train

Dear Sir

We already bought swiss pass via STC, however, we would like to ask for the pass validation. As I'll travelling from Milan to Zurich, to validate swiss pass when train arriving Chiasso, I only write-down date and passport number on the pass, is that enough for ticket validation.
In addition, when i buy ticket at Milan train station do i need to show my swiss pass and pay only for Italy pass (in this case i think pay for Milan to Chiasso, right?) and can use Swiss pass for the rest of Swiss towns, is that correct? . Hope anyone can help me, Thank you very much.