How Far in Advance do you really need to buy tickets for sites?

I would just lock down the Sagrada Familia and the Picasso well in advance.
It has not always been like that but more people than ever are traveling and interested in seeing Europe.

The rest on your list just a couple of days or the day of. The trick is to go real early or real late (like a couple of hours before closing and not between 10 am and 4 pm when the whole world is there; tour groups day trippers etc.

Agree that you need to book Sagrada Familia as far in advance as you can, unless you want to wait in long lines.

We bought our tickets to Casa Batllo in advance also, as we bought tickets for the evening tour/music. It was magical!

I had no problem walking up and buying tickets for immediate entry to Casa Vicens and Sant Pau last week. How long that situation will continue I cannot say, because the other sights you listed seem to be worse than they were in 2016. I would not show up at any of the others without a pre-purchased ticket, but you probably don't have to rush to buy those tickets today. How much time you have in the city is a significant factor. If you have barely enough time to see all your personal "musts" if the entry times fit together perfectly, that's a lot riskier than if you have a longer stay and can take more or less any time period available for the toughest sights.

Edited to add: A complication on a short visit is that you won't have much extra time, so you'll be tempted to book sights back to back. But that requires guessing how much time to allow at each sight and for walking between them. That's quite a challenge. I encourage you to book a first-entry-time ticket for one of the tough sights each day so you at least don't have to worry about getting to those places on time. But those tickets might be the first to sell out.

Two weeks ago it was possible to buy Park Guell and Palace of Catalan Music tickets a few days in advance. And Casa Vincens and Sant Pau had tickets available at the door with no problem.

If you know what dates you want to visit each attraction, I would say ASAP so that you have a better choice of entry time and discounts if they offer them.

Back from a trip to Barcelona mid May 2019. Similar to you I found pre planning all days stressful. Sagrada Familia and possibly Palace of Catalan Music are a must for pre book in advance. I booked 6 weeks in advance. Park Guell for select times would require 4-6 weeks in advance for non peak travel days. No, I-2 days advance sells available. I did enjoy walking the non pay portion of the park. IMO, the pay portion park sites are overrated for the current price (10 E/pp) and time involved. Excellent examples of the tile work can be seen at Palace of Catalan Music and Casa Vicens. I booked La Padrera the night before for first opening timed visit. Picasso Museum could have been booked a few days ahead. Suggest you check the web sites current availability, tickets could be gone at this time. Note, visiting these sites did significantly increase our travel budget. We did the offered tours at The Sagrada Familia (excellent) and The Palace of Catalan Music (good).