What factors influence a star's temperature and density? Assuming a relatively even proportion of mass and radius, a 0.25 solar mass and radius star would have a density of 22.5003 g/cm³, or about 16 times our Sun's density.
However, there is obviously a range of densities that this sized red dwarf could have.  One of the primary factors is its temperature, with cooler meaning a higher density.
My question is three-pronged: what would increase or decrease this sized star's temperature, how would this impact its density, and what factors other than temperature would increase or decrease this star's density?
For main sequence stars there is not much choice. If you pick mass, then all the rest is determined; any variation you see is due to different chemistry (thus fractions of different elements).
For the main sequence one generally assumes that $L \propto M^{3.5}$ and $R \propto M^{2/3}$ (and similarily the density is given for a given mass). The Hertzsprung-Russel diagramme is expression of this.
Any stars not on the main sequence, be that the white dwarfs or any giants, are there due to their peculiar chemistry. White dwarfs are degenerate stellar remanents, slowly cooling. The giants have evolved away from main sequence. They have some other fusion process(es) than hydrogen burning in their core so that their luminosity is much bigger than hydrogen-burning stars; at least they have a hydrogen-depeleted and helium-enriched core and started hydrogen-shell burning.