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The mathematics equivalent of the any% speedrun community?

Yes. Prominent example is the axiom of choice. There was a LOT of discussion about it by mathematicians. https://plato.stanford.edu/entries/axiom-choice/

For instance, with it, you can double the volume of a body by looking at it from a weird angle.

That's a very misleading description of Banach-Tarski. You need to break up the body into a few (very weird) pieces and maneuver them (via only rigid motions - rotations and translations).

Paraconsistent logic looks into logical systems that allow contradictions.

A lot of set-theory research is looking into the consequences of various axiomatic assumptions.

It’s not sub-field. It’s a technique. You relax or change axioms defining your thing and see what happens.