Gödel's incompleteness theorems show that even logic and mathematics require certain assumptions (I think?). What would a world look like if logic and math as we know it did not hold?

I can't address everything in your post, so I'll just try to answer a couple of possible of points of confusion:

Gödel's incompleteness theorems show that even logic and mathematics require certain assumptions (I think?)

I wouldn't put it that way. To state the theorems very crudely, the first incompleteness theorem says something like the following: if A is a consistent collection of axioms (meeting various technical conditions), you can find a sentence S that is neither provable nor refutable from A. In particular, the second incompleteness theorem says that the sentence "A is consistent" is neither provable nor refutable from A (again, given that A is consistent and meets other technical conditions.)

It's not clear to me that either of those statements say anything about whether "math requires axioms."

From what I gather, Principia Mathematica tried and failed to prove math from pure logic. It actually needed three axioms...

That's right, but let's separate two issues. There are at least two ways PM could fail to show that all of math can be proven from "pure logic."

One way it could fail would be if the axioms of the system weren't "purely logical." Whether an axiom is "purely logical" or not depends on your philosophical views concern what "logic" is (see, for example: https://johnmacfarlane.net/FKL-offprint.pdf).

The other way it could fail would be if the system is incomplete - that is, if there's a true mathematical statement that isn't provable from PM.

Notice that these are separate things. To accuse PM of not having purely logical axioms is merely a matter of looking at the axioms and judging whether they are of a "purely logical" character, whereas to say PM is incomplete is a matter of there being a true statement not provable from PM. You don't need to know whether PM is incomplete to decide whether the axioms are 'purely logical', and you don't need to decide whether the axioms are 'purely logical' to know whether PM is incomplete.

The reason I bring this up is because you seem to be conflating the Godel's incompleteness theorems with the issue of whether PM has non-logical assumptions. The issue about whether PM's assumptions were "purely logical" was discussed quite extensively before Godel's theorems (see: https://en.wikipedia.org/wiki/Axiom_of_reducibility). The incompleteness theorems don't say anything about that issue. Instead, what they show is that PM can't possibly hope to prove every truth of mathematics, regardless of whether its axioms are considered "purely logical" or not.