Thank you for your request.
1.The price of the Swiss Half Fare Card for 1 month 120 Francs, when you buy a ticket you must, when you subscribe the passengers with the name and birthdate choose the reduction: Half Fare Card.
All informations for the Swiss Half Fare Card here: Swiss Half Fare Card – Switzerland at half the price | SBB
2.The Day Pass is valid for one Day from 00:00 am till 5:00 am next morning, and the start prices are from 75.00 francs and higher, you have also the option of the Saver Day Pass which is cheaper but a limited offer all informations for this two products you have here: Saver Day Pass – book earlier, travel cheaper | SBB, Day Pass for the Half Fare travelcard | SBB.
Under this link you find also the validity map of the Day Passes.
3. No for the train from Zürich to Paris, you will have not any discount with the Swiss Half Fare Card.
4. If you are travelling a lot this 4 Days then is worth to think about buying the Swiss Travel Pass, all Informations here: Swiss Travel Pass – the All-in-one-Ticket for Switzerland | SBB
I wish you a lovely Day,
Kind Regards MelanieD