The problem of the most visited point.

Solution 1:

I think it' strictly linked to the self-avoiding path problem (which is slightly different as the circuit doesn't necessary ends where it started); there is also a detailed study , and a partial solution to what you called $N_n(x_{\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor})$, all circuits which visit the origin. Anyway until now, any solution has been found. It can be an idea to consider a circuit that starts from $x_{ab}$ and arrives to $x_{cd}$, and then consider the reverse paths (from $x_{cd}$ to $x_{ab}$), remembering not to count the reverse paths which intersect the first one, as to create an Hamilton path.