What are the conditions under which one would make $c\sim 0$ or $b\sim 0$ for $\vec{f}(v) \sim - (b v + c v^2) \hat{v}$?

The linear coefficient $b$ depends on viscosity of a medium; for a non-viscous medium it can neglected. It also depends on the object's linear size, so one can also neglect it for a small object. The coefficient $c$ depends on cross sectional area of the object, and the density of the medium, so can be neglected for rarified objects and small objects. Whether or not you can neglect either term depends as well on the velocity; at high velocity the quadratic term becomes relatively more important.