(CW because I don't say anything "mathematical" [i.e. answers questions 1 and 2]; so the below is a subjective and biased opinion on question 3.)
Would it be publishable of you solve the problem?
Given the state of the publishing industry nowadays, hardly anything is "not publishable". So I assume you mean publication in a suitably serious journal. That is then tied to the next question...
Would it be worthwhile?
First you should ask yourself why was this question asked in the first place. So what does a separation of variables give you? One big advantage is in the construction of explicit solutions. In 1D the problem of computing the spectrum of perturbations of the Laplacian is immensely more tractable than in higher dimensions in general, and the associated generalized eigenfunctions can be obtained by solving ODEs. This makes it much easier to actually go in and by hand do the Fourier decomposition.
As an example, on an arbitrary compact Riemannian three manifold we know we can decompose any function $f$ into eigenfunctions of the Laplacian. But if you give me a three manifold and a function and ask me to compute the Fourier coefficients and the eigenfunctions, I can just through my hand up in the air and say: "no can do, unless you want numerical results". If you work on the torus, however, things can be done explicitly because of the nice separation of variables.
So separation of variables for the Laplace and Helmholtz equations give us a method of explicitly solving the Poisson equation and the Cauchy problem for the wave equation using this method .... on Euclidean spaces, however, these can also be done by convolving with the explicit integral kernel. Before computers are widely used, the former has the appearance of giving a larger class of explicit solutions.
In terms of PDE theory, we have moved quite far from "separation of variables for Laplace/Helmholtz". Part of it is because the study of linear equations with constant coefficients are by now considered largely uninteresting: modern studies focus much more on PDEs with variable coefficients, nonlinearities, or both. (I should note that the importance of considering variable coefficients, in terms of equations with potentials, was expected even in the 40s.) For nonlinear equations, in particular, the classical separation of variable ansatz becomes of rather limited use.
This is not to say that people in pure PDEs don't care: just that you have to realign your paper and your goals to the current interest. So in terms of variable coefficients it may be more worthwhile to study separation of variables on Riemannian or pseudo-Riemannian manifolds, with respect to their Laplace-Beltrami operators, rather than restricting yourself to higher dimensional Euclidean spaces. Or you can dive deeper in some of the more modern developments that are, at some point, connected to separation of variables techniques. For example you can study symmetry properties of differential equations (see Olver's book for a starting point). Or you can study the problems from the point of view of "generalised-" or "functional-" separation of variables.
Another possibility is of course to consider the problem less from the pure point of view but more in terms of its (possibly computational or engineering) applications. For that I am the wrong person to ask. But based on the publications of various handbooks on linear PDEs, there does appear to still be substantial interest in methods of solving linear PDEs from the engineering community.