Hello Doqueayee99
It is not relevant from where you book your tickets, SBB or Trenitalia. The trains are jointly operated by both train companies, and you cannot choose to ride with one or the other. The trains in Switzerland are operated by SBB staff, once they cross the border at Chiasso, the crews will change and the Italian rail staff takes over.
When travelling with a Swiss Travel Pass (STP), or a GA Travel Card, customers can declare it as a Discount cards when purchasing tickets. By doing so, they will only pay for the Italian stretch, the Swiss stretch is not charged but the seat reservation is valid from Zurich to Milano.
It is correct that your child travels for free on the EC trains, given it travels on your lap. If you want an extra seat, you will need a child ticket. The seat reservation is included for everyone with a valid ticket.
Please note (for other users reading this answer),
- only Swiss Travel Pass or GA Travel Cards can be declared as Discounts cards. Saver Day Passes and other Day Passes do not count as Discount cards; with such tickets, one has to purchase two separate tickets, as suggested by @Danhiel .
- per person, one child is permitted to travel on the lap.
Kind regards,
Roland