Alhambra - which entrance

I think you will be fine having them on your phone and not printed. We didn't even show our tickets when we went last week, they just scanned our passport to enter and that was it. Our tour guide said that the tickets are only in case the passport doesn't work. (We are US citizens and had US passports FWIW.)

Also, I do not think the gate matters for this, we went last week and you do not show your tickets at all when you enter through the Justice Gate, you show them to enter the area around the Tower and again to enter the Nazrid Palaces.

If you want to visit the Nasrid Palace first, then I recommend entering through the Justice Gate. It's close to the Nasrid Palace, and it's not as busy as the other entrance.

The Alhambra and access to it—ticketed and/or physical— can be kind of confusing prior to seeing it in person. The Justice Gate is a shortcut when you're walking up the hill to (or down the hill from) the larger, total Alhambra complex— it's a gate through the city wall. All the elements, buildings, and sites, are then inside the city walls, and have their own entrances, at which you'll present tickets or passports.

Thank you everyone for your responses. That was super helpful. We had no problem going through the justice gate as there is no ticket required at this point. We also had no problem having the QR code on our phone when we went through the entry point to the Nazaries palace.