I stayed in a place across the street from the Palau de la Musica, so I got to know a few places nearby.
Instead of the metro, with a change, take the bus to Placa de Catalunya. It's comfortable, pleasant and drops you near the southeast corner of the plaza, so it's probably not any farther from your hotel than the Urquinaona metro. I also found it confusing to figure which exit was best from that station and once at street level, hard to figure out which way I needed to walk. Buses are frequent, it's nice to look out the windows as you enter the city. There's a ticket agent at the bus stop at the airport, just pay, grab your ticket and hop on the bus. Luggage goes on the bus with you.
I haven't been to Park Guell. I followed advice I got here and chose the first entry time to the Sagrada Familia. Get there 15-30 minutes in advance to be one of the first to enter the church. You'll have quiet time to absorb the grandeur and beauty and take some mostly people-less photos. An hour or two later, the din is likely to be deafening - people talk loudly (it's not a functioning church - services are held in a separate chapel underneath) and there's continual construction going on, so you'll probably hear hammers, drills, etc. From here it's an easy 10-15 minute walk along av. de Gaudi (lined with cafes if you want to stop for lunch or a snack) to Sant Pau. You can visit with a guided tour now - when I went there weren't any. I was quite happy to wander on my own with the printed info provided. BTW every building you can enter (some are closed to the public) has lots of toilets - after all, it was a hospital ☺
You'll probably have to buy tickets in advance for the Palau de la Musica - it's only by tour. Don't skip the cafeteria - it's as lovely as the rest of the building. BTW prices are reasonable if you want a drink or snack. There's a good bar/restaurant across the street from the entrance, Tosca. My favorite was Orio, which serves Basque tapas. There's one across the street from the entrance to the Santa Caterina market. Order one glass of cider, just to watch the waiter pour it. Another place not to miss in the "neighborhood" is a hole-in-the-wall churros shop. It's on via Laietana a block or two (really can't remember) down from the Palau, on the left if you're walking down (toward the sea). I was ready for a snack, but I couldn't find the churros place I'd been told about and I was too tired anyway. I resigned myself to going back to my room picking up a snack at a kiosk when I just happened to look in a window as I was walking along and there was a man frying churros in a big vat right in front of me! The best churros ever!!! One helping - in a paper cone, simply sprinkled with sugar - was €1 or 1.50 and plenty for two people. The hard part was waiting for them to cool off enough to start eating them. They also make potato chips. One more place - Elsa y Fred, less than a 10-minute walk from the Palau, nice atmosphere, excellent food, reasonable prices.
There's a rooftop tour of the Church of Santa Maria del Mar that's quite good. The price includes entry to the church, so you can explore either before or after the tour.
Sadly, it appears that Casa Lleo Morera is closed - there's a virtual visit on their website but no info on actual visits.
