I can show the first identity $R(n,0) = f_{n+1,n+1}(0)$, as a consequence of the more general identity

$$ R(n,q) = \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)\tag{1}\label{1}$$
for $n,q \geq 0$.  Indeed, note that $g_{n+1,n+1} \equiv 0$ and thus $f_{n+1,n+1}(0) = f_{n+1,n}(n+1)$, so $R(n,0) = f_{n+1,n+1}(0)$ is basically a special case of \eqref{1}.

One can prove \eqref{1} by induction on $n$.  For $n=0$ we have
$$ R(0,q) = 1 = \frac{1}{(q+1)!} f_{q+1,0}(1)$$
by hypothesis.  For $n>0$, we see from Taylor expansion (and noting from induction that $f_{n+q+1,n}$ is a polynomial of degree $q+1$) that
$$ \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)  = \sum_{j=0}^{q+1} \frac{1}{(q+1)! j!} f_{n+q+1,n}^{(j)}(n).$$
But from the recursive definition of $f_{n,\ell}$ and the fundamental theorem of calculus one has
$$ f_{n,\ell}'(x) = (n-\ell)^2 f_{n-1,\ell}(x)$$
and hence on iterating
$$ f_{n+q+1,n}^{(j)}(x) = \left(\frac{(q+1)!}{(q+1-j)!}\right)^2 f_{n+q+1-j,n}(x)$$
so that
$$ \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)  = \sum_{j=0}^{q+1} \frac{(q+1)!}{(q+1-j)! (q+1-j)! j!} f_{n+q+1-j,n}(n).$$
But from the recursive definition of $f_{n,\ell}$ and the induction hypothesis we have
$$ f_{n+q+1-j,n}(n) = f_{n+q+1-j,n-1}(n) = (q+2-j)! R(n-1,q+1-j)$$ 
and hence
$$ \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)  = \sum_{j=0}^{q+1} \frac{(q+1)!}{(q+1-j)! j!} (q+2-j) R(n-1,q+1-j).$$
Replacing $j$ by $q+1-j$ we obtain
$$ \frac{1}{(q+1)!} f_{n+q+1,n}(n+1)  = \sum_{j=0}^{q+1} \binom{q+1}{j} (j+1) R(n-1,j)$$
recovering \eqref{1} by the recursive definition of $R$.

I discovered \eqref{1} by working backwards from the desired identity $R(n,0) = f_{n+1,n+1}(0)$ to try to write $R(n,1) = \frac{1}{2} (R(n+1,0) - R(n,0))$ and then $R(n,2) = \frac{1}{3} (R(n+1,1) - R(n,0) - 4 R(n,1))$ in terms of the $f_{n,\ell}$ in as simple a fashion as possible (using the ODE form $f_{n,\ell}' = (n-\ell)^2 f_{n-1,\ell}$, $f_{n,\ell}(\ell) = f_{n,\ell-1}(\ell)$ of the recursion for $f$ to eliminate the role of the indefinite integral $g$), at which point the general pattern became clear.  Presumably a similar technique, using the multilinearity and symmetry properties of the permanent to express $R(n,1)$, then $R(n,2)$, etc., as a suitable permanent starting from the hypothesis $R(n,0) = a(n+1)$, will then give a similar identity relating $R(n,q)$ with the permanent of a suitable matrix which can then be established by induction to complete the proof, but I will leave this for someone else to try. EDIT: actually it may he easier to try to discover how $f_{n,\ell}(x)$ can be expressed neatly as a permanent, as it has a simpler recursion.