As an atheist I have an obligation to finish reading it all (still going through, and taking notes, probably having to revisit), but I am not sure how many (christian) believers will feel the same.
Priests will read it and then talk to their congregations about it on Sundays throughout the year, if not explicitly, then in how it shapes their homilies.
In all my life of being Catholic (I’ll turn 50 this year), I’ve heard less than 5 homilies-sermons that amounted, in whole or part, to a reflection on a papal encyclical. Over time there may be juicy papal quotes that make it into Sunday preaching, but that’s about it.
Instead, priests tend to focus on the readings for that Sunday’s Mass and more general themes.
That being said, I hope many priests do read an encyclical any time a pope publishes one, but they’re very, very busy most days and weeks, so whether any one priest will commit time to reading a particular encyclical, old and dusty or hot off the presses, will depend on a lot of factors that are as varied as their individual circumstances and personalities.
In theory only and all Catholics recognize the authority of the pope. In practice it’s a mess as far as I understand, with a bunch of American catholic groups who rejected church reforms that happened during the 20th century, resulting in people calling themselves catholic who do not actually believe that the Catholic Church has authority over their religion.
Add to that the fact that the pope has a cultural influence that goes further than only the catholic audience (lots of Protestant see the pope as important even if that’s not something dictated by Protestantism, a bunch of not really religious people see him as a sort of spiritual leader, etc)
I'm not aware of any Protestants that see the Pope as important except in a very negative way - that's practically the defining feature of Protestantism and one of the few things all the Protestant denominations have in common, whether "low church" or "high church".
Heck, it's a struggle to convince many of them that Catholics are Christian at all, and "the Pope is the antichrist" used to be a normal, mainstream comment in American newspapers.
It is somewhat a piece of irony that the Pope generally holds a more favorable reputation in the minds of the irreligious in America than the religious. Even the average Catholic likely does not have as much respect for the Pope as some of the commenters here.
This is not very common in Continental European protestants, it is more something American protestants inherited from British protestants, that had some peculiar reasons to turn the dial on papal hate to the 11.
Any serious Protestant (mainline and knowledgeable) listen and give weight when the Pope speaks. They would certainly refuse the Pope authority and inerrancy ex Cathedra but not necessarily disagree with him. Theologically there's much less division between Protestants and Catholics today than in XVI. A large share of disagreeements are due to residual historical animosity.
Every mainline Protestant denomination was basically founded on the idea that they shouldn’t have to listen to the Pope at all. And that was before Papal infallibility was enshrined, which actually didn’t take place until the late 1800s.
In America, anti-Catholic sentiment was extremely strong until relatively recently, and then only because religiosity (and thus the reason for it) has declined. All the theological division still exists, it’s just less striking in a world that’s much more irreligious and in countries where vastly different religions (Muslim or Hindu) are present now in real numbers.
Practically all the pro-Pope sentiment I’ve seen in my lifetime has been from various flavors of atheists, agnostics, and other areligious types. Catholics themselves generally hold more respect for the office than the person, and Protestants are almost uniformly negative on both.
Vatican II started a major and ongoing reconciliation process leading to things like the "Joint Declaration on the Doctrine of Justification" (1999, quietly resolving the core issues of the reformation) and Pope Francis commemorating the 500th anniversary of the reformation at a Lutheran church in Sweden (2016):
I think attitudes vary regionally and by congregation and an ecumenical focus doesn't necessarily translate to a positive perception of the pope but it can and not just all that recently.
