Is it worth buying tickets on the Trenitalia website here? Do the website tickets automatically give us a reservation if needed or is this something we still have to get at the train station once there? If buying tickets there, is 1- 2 days beforehand enough time to get a reserved seat at a specific time or will we risk having to travel at other times?
For trains that require reservations, the reservation is included in the price of the ticket whether you buy at a station or book online. Booking well in advance gives you the best chance of getting Amica fares (20% discount), but buying tickets 1-2 days ahead of travel will work just fine for virtually any train at any time of day. And you might still be able to get Amica fares.
If you are taking an overnight train but in advance - those trains sell out quickly.