First of all,mechanical energy isn't conserved when $\vec{F_{nc}}$ is there in the system unless it does zero work.
Coming to swapping differentials, isn't that like this: $$adx=\frac{dv}{dt}dx$$ Multiplying and dividing by $dt$ $$adx=\frac{dv}{dt}\frac{dx}{dt}dt$$ $$ = \left( \frac{dv}{dt} dt \right) \frac{dx}{dt}$$ $$ = \frac{dx}{dt} dv$$ Now, $$adx=vdv$$
The last integral $\int_C (a_x \hat{i}_x + a_y \hat{i}_y + a_z \hat{i}_z) \cdot d\vec{r}$ can be carried out if $d\vec{r}$ is written as: $$dx\hat{i}_x+dy\hat{i}_y+dz \hat{i}_z$$ Carrying out the dot product with $\vec{a}$, it will be given as: $$a_{x}dx+a_{y}dy+a_{z}dz$$
It's easy to swap differential here as we get $a_{x}dx=v_{x}dv_{x}$, $a_{y}dy=v_{y}dv_{y}$ and $a_{z}dx=v_{z}dv_{z}$.