Second question
Two comments here. First, the variation of $\rho$ with $q_a$ is not $\mathrm d\rho/\mathrm d q_a$ but $\partial \rho/\partial q_a$. Second, to calculate $\mathrm d\rho/\mathrm d q_a$ start with the differential $\mathrm d\rho$: $$ \mathrm d\rho = \frac{\partial\rho}{\partial t}\mathrm d t + \frac{\partial\rho}{\partial q_a}\mathrm d q_a + \cdots $$ then divide by $\mathrm d q_a$: $$ \frac{\mathrm d\rho}{\mathrm d q_a} = \frac{\partial\rho}{\partial t}\frac{1}{\dot q_a} + \frac{\partial\rho}{\partial q_a} $$