which homogeneous polynomials split into linear factors?

1) This is the Chow variety of degree $n$ zero cycles in $\mathbb{P}^{d-1}$.

2) Yes, this collection of polynomials can be bundled together into the Brill form or covariant.

3) Rather explicit descriptions of the Brill equations can be found in the book by Gelfand, Kapranov and Zelevinsky on resultants. There is also a paper by Rota and Stein. But first check out Emmanuel Briand's page and in particular the articles "Covariants decomposing on totally decomposable forms" and "Brill's equations for the subvariety of factorizable forms" and if you read French (or German) the translation of the original article by Gordan (respectively the article itself).

As an aside, analogues of the Brill equations for the variety of forms which are powers of forms of degree dividing $n$ have been given recently in my paper with Chipalkatti "On Hilbert covariants".

There are Brill's equations. Look for them at the book by Gelfand, Kapranov, and Zelevinski. In general Brill's equations do not generate the ideal of totally decomposable polynomials, see this paper.

The questions 1. to 3. can be answered by using the theory of Gröbner bases. Let

(1) $f(x_1,\ldots,x_d) = \sum a_{i_1\ldots i_d}\, x^{i_1} \cdots x^{i_d}$

with indeterminate coefficients $a_{i_1\ldots i_d}$.

Now assume there is a factorization

(2) $f(x_1,\ldots,x_d) = \prod_{i=1}^n (b_{i1} x_1 + \cdots + b_{id} x_d)$

again with indeterminate coefficients $b_{ij}$

Multipliying (2) out and equating coefficients of like monomials $x_1^{i_1} \cdots x_d^{i_d}$ in (1) and (2) gives a set of polynomials

(3) $G_\nu(\ldots,a_{i_1\ldots i_d},\ldots, b_{ij}, \ldots) = 0$

Eliminating from the $G_\nu$ all the $b_{ij}$ by Gröbner basis methods gives a set of equations $F_\mu(\cdots,a_{i_1\ldots i_d},\cdots) = 0$. These describe $S$ as an algebraic variety.