Cantor's work in the early 20th century was on infinite objects in the context of set theory. That work is also fairly well-understood but starts to get a bit esoteric. More recent research goes further in the same direction, and then you start getting the point where you can't prove whether a certain type of infinity exists or not, because the existence of an infinity that large would prove that set theory is consistent.
So yes, research is still ongoing into the nature of infinity, but it's the nature of a different infinity.
The "-1/12" "mystery" is well settled math (but mathematicians are always looking for more elegant or general perspectives on things), and (standard-issue) sloppy notation in physics.
The only "mystery" is why people who should know better intentionally use incorrect terminology to dazzle lay-people, creating a mystery where there is none.
I'm no mathematician either, so I'll stop there before I embarrass myself. The most accessible resource I found was this VSauce video [1].
^1 Basically large cardinal axioms give you additional types of infinity.
https://en.wikipedia.org/wiki/Continuum_hypothesis
I remember learning about them in the philosophy of maths, but have no idea if they have any consequences in Physics. And like /u/lasfter I'll stop there before I embarrass myself.
I am not a mathematician, but my understanding is that a considerable amount of research into how to properly address /use infinity in math was ongoing as recently as the early 20th century. Not that I disagree with your overall point though.