Hello Dinieharis
If it is for this trip only, then I would not bother using the app. Simply go to the station office in Lucerne and let them know that you hold the STP and wish to purchase a class upgrade. To purchase the upgrades online, it would require too much administrative effort, see my instructions below.
To answer your questions:
Yes, upgrades can be purchased in advance, at any time.
You are eligible to purchase class upgrades at half fare.
You can purchase the upgrades using the SBB Mobile App, but you have to register any traveller as co-passenger in your app. The upgrade tickets must be issued for each individual traveller (they must bear the name of each traveller, and each traveller must carry their physical passport).
If you travel with children, and you hold the Swiss Family Card, you do not need to purchase class upgrades for the children as they are covered by your STP and your upgrade ticket.
Travelling with 13 people is quite an undertaken. It might be useful to make a group seat reservation for some of your trips. It needs a preplanning and a prearrangement; see the station office for that.
Kind regards,
Roland