Given a standard advection equation, we write the update as $$ q_i^{n+1}=q_i^n+\frac{\Delta t}{\Delta x}\left(F_i^{n+1/2}-F_{i+1}^{n+1/2}\right) $$ with $F_i^{n+1/2}=F\left(q_i^n,\,q_i^*\right)$ and $q^*_i$ a predicted value, other variables take their normal meaning.
It is possible to reconstruct what numeric value $F_i^{n+1/2}$ possessed if I know $q^n_i$ and $q^{n+1}_i$? Or is it once I deallocate the array F, that information is lost to me?
You could solve a linear system in terms of the $F_{i}^{n+1/2}$.
I am somewhat confused by this question. Why are you evaluating the fluxes at a midpoint of the time step instead of evaluating them at $t = t_{n}$ or $t = t_{n+1}?
From what I have seen, fluxes are generally using data at the same time step to produce a value $q^{*}$ at the interface of two elements, where $q^{*}$ is the solution to the Riemann problem generated at the interface. This $q^{*}$ is then used as the variable to evaluate the flux function at, at this interface.
For nonlinear problems, finding $q^{*}$ is difficult and many people opt to use approximate solutions to $F(q^{*})$ via linearization about $\hat{q}$ which is usually some function that generates some sort of average of the right and left values of $q$ at the interface, or other fancier approaches to approximating the correct flux value at the interface.
Whether you are aiming for an explicit or implicit scheme, the approach is the same (at least in my experience), even if the value $q^{*}$ ends up helping create a coupled system in the implicit scheme case. I will recognize that I still have plenty to learn, so I definitely don't know everything about how to approach these problems, just the ways I have been taught.
