For example, in the same fashion that medical professionals might do research on cancer, does there exist comparable positions in philosophy where one "researches," say, Free Will Determinism?
Yes, for example Oxford's philosophy faculty includes professors of
ancient greek philosophy
philosophy of mind
philosophy of language
philosophy of physics
philosophy of mathematics
There are philosophy faculty who specialize in research on a particular topic (see @henning's answer). However, what I think is rare in philosophy and other Arts and Humanities disciplines, it research only positions. Where as in sciences there are purely research, commonly at the postdoctoral level, but also occasionally as full-time permanent employees (e.g. PIs research institutes or staff scientists). Most academic philosophers I know are also teachers to one sort or another, even at the Postdoc level.