The Euler Lagrange equation $\frac{\partial F}{\partial q}-\frac{d \frac{\partial F}{\partial \dot{q}}}{d t}=0$ can also be put in the form $\frac{\partial F}{\partial t}-\frac{d (F- \dot{q}\frac{\partial F}{\partial \dot{q}})}{d t}=0$,
How is the second form of the equation arrived at mathematically, and does equating them and rearranging (as above) lead to any simplifications of the equations? I'm probably missing some elementary rearrangement.
It looks like you are making some confusion with partial and total derivatives. But the second form writen as:$$\frac{\partial F}{\partial t}-\frac{d }{d t}[F-\dot q \frac{\partial F}{\partial \dot q}]=0,$$
is nothing but the conservation of energy, since $\dot q \frac{\partial F}{\partial \dot q}-F$ is defined as the Hamiltonian of the system, or by other words the energy.
The way to derive this form is only to expand the total derivative of the function (in physics: Langrangian) $F$ in order to it's explicit dependencies in $q$, $\dot q$ and $t$ and make use of the Euler-Langrange equations to re-write the term that involves the partial derivative in order to $q$.