Caveat: My recent travels to Spain were in 2016 and 2019, pre-pandemic. Visitor levels will be lower this year, but if there are places still operating with reduced capacity, that could affect ticket availability. I have no idea whether the areas you're visiting have lifted all their capacity controls.
Casa Mila, Casa Batlio, Parc Guell, Palau de Musica (guided tour and perhaps a concert): Yes. This may not need to be done way ahead of time, but walking up without a ticket may put you in a very, very long line, and you might encounter a sellout. (Both the houses were totally packed at the time of my visits; they were letting people in only as others left.) There are no Parc Guell tickets sold at the park itself. These are not cheap tickets, so I wouldn't be in a rush to buy them, given COVID uncertainty. I'd do fake ticket purchases online every few days to be sure my preferred dates were not selling out. You may well be able to wait until just before you leave home. However, I don't know what sort of tourist spike Barcelona sees around Easter. Is that when you're traveling?
Recinte Modernista de Sant Pau: I assume No. I've simply walked up and bought my ticket twice, most recently in May 2019. I've read nothing to suggest that has changed.
Barcelona Cathedral: I assume No. Not a problem for me in May 2016. Check your guidebook; I think the cathedral is free for part of the day. Check for info on dress code.
Monserrat: Don't know.
Sevilla Cathedral: Very long ticket line stretches across the uncovered plaza. It's at least as bad as the situation at the Alcazar, I think. Rick explains how to avoid the cathedral line by buying a combo ticket at the Church of El Salvador. Worked for me in 2019: Bought ticket, saw church and walked right over to cathedral. Line-minder at cathedral waved me straight over to the entry door.
Seville flamenco show: No experience with this, but I'd expect to buy performance tickets at least a bit ahead of time.
La Mezquita: Don't remember buying this ticket early but may have done so a day or so in advance. I believe the site takes a mid-day break. It's another for which I'd monitor ticket availability online (if available) to be sure the time of day I wanted wasn't close to selling out.
Toledo: Cathedral: Not sure. I wasn't able to get a ticket for immediate entry here in May 2016; I had to return an hour or so later. However, it was a holiday weekend; I suspect the cathedral isn't always that busy. But if your visit is around Easter, that might make quite a difference.
Madrid Royal Palace: Not sure. The palace was crowded at the time of my May 2016 visit, but I don't remember much of a delay at the ticket window. There's heavy visitation by Spaniards, so holidays are probably busier than average.
Prado/Thyssen/Reina Sofia: I assume No. I basically walked right in at all of those during May 2016. The Prado seems to get a lot more visitors than the others, but it is huge, so people are spread out. I saw clumps of people in front of only two pictures: Las Meninas and Bosch's Garden of Earthly Delights. It seems that the only thing people want to see at the Reina Sofia is Guernica. I think there may be a combo ticket that save a bit of money if you want to see all three; I haven't verified that, however.
Sorolla Museum: No experience here, but I'd be very surprised if advance tickets are necessary unless there's some sort of high-demand special exhibition running.
Madrid flamenco show: No experience, but I'd expect to prebook this performance.