Barcelona - a very short stay: would love feedback on our itinerary

You definitely need to reserve timed tickets ahead of time for Sagrada Familia and Casa Batllo. Plan on 60-90 minutes inside each site. Try to reserve Sagrada Familia for opening hour at 9am because the security check and entry lines quickly build and you don’t want to waste precious time standing outside in lines as the heat of the day increases. We didn’t need pre-purchased tickets for Casa Vicens and it wasn’t crowded so that worked out fine.

Personally, I would skip Park Guell and instead visit the fantastic complex of Montaner’s architectural masterpieces at the Hospital de Sant Pau. It’s much larger and more impressive than Park Guell, and you can go inside the buildings. It is somewhat near Sagrada Familia, so visiting it after the cathedral makes sense logistically. No need to pre-reserve tickets for Hospital de Sant Pau.

I think your plan for day one works. For shopping, there is an “El Corte Ingles” shopping center right at Plaça de Catalunya where you’ll be staying.

During the evening of Day 1, I highly recommend seeing the interior of Palau de la Musica Catalana— another Barcelona Modernista masterpiece.
If you go to an evening Flamenco or other performance there, you can also see the magnificent stained glass windows and details of the concert hall at sunset before the show begins. Palau de la Musica Catalana is near the Gothic quarter.

Have fun in Barcelona!

To me Casa Vicens is a lot less interesting than the other Gaudi sites you mentioned (and also less interesting than Casa Mila/La Pedrera). Only a very few restored rooms were open as of 2019. Your trip is very short and Casa Vincens isn't particularly conveniently located. I'd skip it.

I agree that both Recinte Modernista de Sant Pau and the Palau de la Musica Catalana are great. I think you have too many possibilities for what amounts to just one full, we-hope-not-jetlagged day in Barcelona. However, the Palau de la Musica Catalana is worth seeing even if there's only time for the fabulous exterior. You should be able to swing by there on the way to or from the Barri Gotic. I'd be too sleepy and jetlagged on arrival day to want to go to a performance of any sort.

I'd plan a rock-bottom minimum of 90 minutes at each Gaudi site. Those are expensive tickets, and those places have a lot to see. There are good audio guides, and the Sagrada Familia museum is very interesting. They're also crowded, which is likely to slow you down to some degree. As already mentioned, your schedule also needs to allow for possibly not getting inside until later than the time on your tickets. And then there's travel time from site to site and lunch.

In one wonderful day, we went to Sagrada Familia with reserved tickets at opening, walked to Recinte Modernista San Pau, had lunch nearby, then took a bus up to Parc Guell. Sagrada Familia and San Pau were fabulous. Parc Guell, while interesting, was over crowded and, for us, missable. Staying near Placa de Catalunya is perfect. Enjoy!

1. No, stick to two (max three) sites on a single day -assuming you've got the whole day. My choice?... ParkGüell or Recinte Modernista Sant Pau > SagFam > La Pedrera in the evening
2. I don't get the question: there are over two dozen Modernist sites to see in Barcelona, not just "three casas".
3. You need to prebook in advance, for some sites (ie, SagFam), tickets are only sold online. You choose your time slot and visit at that hour, no waiting in queues

For your teen: there are literally hundreds of shops aimed at teens everywhere, but since you're just staying a day and a half... these stretches in the centre (and the adjacent narrow streets) are full of shops she'll love: https://www.scribblemaps.com/maps/view/BCN-Teen-shopping-centre/GoV8X0IiZd

Enjoy!

What a great feedback and advice! Really appreciate you all sharing your thoughts on the logistics, sites to see / skip, tickets, etc., much appreciated!
I have added Recinte Modernista de Sant Pau (in place of Park Guel) and Palau de la Musica Catalana (and will look into shows / concerts there, but might be too much). Yeah, I know, we will be very jet-legged, but, hey, we don't get to visit Spain that often to allow too much rest :) We'll rest in trains when travelling between cities.