With regards to your second question, I think it depends on what properties you want your higher dimensional node to keep. This I suppose answers the (slightly different) question of "what is a higher dimensional analog of a node?" (and for simplicity, as already pointed out, I'll work over an algebraically closed field).

The singularity you describe is an example of *simple normal crossings hypersurface*. In many cases, it is a perfectly good generalization of a node. In other cases, you might want to allow more singularities or (not allow your simple normal crossings hypersurface at all). Your higher dimensional analogs of nodes might be hypersurfaces, or have isolated singularities, depending on context.

For example, one place where nodal curves show up is when doing generic projections of smooth curves in $\mathbb{P}^n$ to $\mathbb{P}^2$ (see Hartshorne, Chapter IV, Section 3). In general, you can take a $d$ dimensional projective variety and generically project it to $\mathbb{P}^{d+1}$. Such generic projections will be something called ''seminormal'' (see a paper by Greco and Traverso), and they will also be Gorenstein (they are hypersurfaces). These conditions are equivalent to being a node in dimension 1. On the other hand, these conditions on a singularity are not equivalent to the singularity being a generic projection in general. Rob Lazarsfeld also discusses generic projection singularities a little bit in his ''Positivity'' book.

Another place that nodal curves show up is in the usual compactifications of moduli spaces of curves. If you look at higher dimensional varieties, then presumably the correct generalization of a node is then something called ``semi-log canonical singularities''. This is again a distinct notion from these semi-log canonical hypersurfaces, see for example Rob Lazarfeld's book and also the dissertation of Davis Doherty (University of Washington, 2006).