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Pius XII, Humani Generis:

For these reasons the Teaching Authority of the Church does not forbid that, in conformity with the present state of human sciences and sacred theology, research and discussions, on the part of men experienced in both fields, take place with regard to the doctrine of evolution, in as far as it inquires into the origin of the human body as coming from pre-existent and living matter - for the Catholic faith obliges us to hold that souls are immediately created by God.

This here is one of the most tragic statements in the history of the Magisterium.

First of all, though he doesn't say it, there's a strong implication that the Teaching Authority of the Church should bow to the "present state of human sciences", almost putting theology and the sciences on a par, on the same level.

Secondly, the notion that the human body came from "LIVING" matter is contrary to Sacred Scripture AND the unanimous consensus of the Church Fathers, who all assert unequivocally that Adam was created from the earth or the dust of the earth.  How "earth" or "dust of the earth" can be extrapolated into an "ape" without doing extreme violence to Sacred Scripture is absolutely inconceivable.  Where you could perhaps apply Providentissimus Deus is to say that the dust of the earth refers simply to the various chemicals and organic compounds that are found in abundance in the earth.  If you look at the chemical composition of the human body and that of earth or soil, there's a very close match (I'll dig it up at some point).  But if you hold that when Scripture states that God created Adam from the earth, the "earth" was just an ape ... then all of Genesis becomes a non-historical allegory of some kind replete with errors.

But Pius XII is not teaching that this was the case.  In fact, if you read on, he says that reasons both for and against must be considered, and the conclusions ultimately subjected to the Magisterium.  So all he's doing is opening up some freedom of discussion on the matter.  But that in and of itself was a horrible, horrible mistake.  We have this statement to thanks for the millions of Catholics who now hold that Genesis was merely a fanciful story.  And cassini points out the role that Providentissimus Deus played in this as well.  Where Leo XIII gave them the key, Pius XII unlocked the door ... which the Modernists then flung wide open.  It's similar to Roncalli talking about opening up the windows of the Vatican ... only to allow all the sewage into the Church.

The talk about geocentrism brought to mind a catechism class that I downloaded from SSPX Asia at least 10-12 years ago (relevant clip attached below). Fr. Scott gets upset that anyone would consider geocentrism true.

I think it is an example of why we need to examine the beliefs that we have picked up from the world, especially "proven science".

Couldn't agree more!   With the amount of lies and disinformation that has been getting light since the internet came around, makes me basically question anything "science" claims to be true.  And if some atheists scientist says one thing and Church tradition says another, guess who I'm siding with.  
Another consideration is relevancy.  Does it matter one way or another the way space works or the distant past of the earth?  In my opinion pretty much no, anything that is not working towards the salvation of souls is ancillary at best and totally irrelevant at another end of the spectrum.