Hello RJennyfer
If you and your friends are under 25, you can buy this pass:
https://www.sbb.ch/en/tickets-offers/tickets/day-passes/friends-day-pass-youth.html
Otherwise, you can buy Saver Day Passes, be sure to select the full price:
https://www.sbb.ch/en/tickets-offers/tickets/day-passes/saver-day-pass.html
Can I take unlimited trains with this ticket for the day? (the route would be Varese-Zurich-Bern-Interlaken)
These passes are valid from/to the border station of Cantello-Gaggiolo. In Switzerland unlimited.
How does seat reservation work? Can I just hop on the train and pick a seat?
You can find more information here:
https://community.sbb.ch/blog/23887-demystifying-seat-reservations-on-swiss-trains-a-comprehensive-guide
Do I need to show the ticket and the purchase receipt during inspection?
You must show the ticket(s), everyone must also present a physical passport or identity card (not only photos).
Can I use the pass for other types of transport? Is the ticket valid in Zurich?
Yes, day passes are also valid for public transport in the cities. You can find the area map of validity here (select Saver Day Pass / Special Day Pass):
https://www.sbb.ch/en/tickets-offers/travelcards/ga-travelcard/ga-travelcard-area-validity.html
Kind regards
Danhiel