First, the Gröbner basis is not sparse. I am speaking a little off-the-cuff, but empirically when I ask SAGE for the Gröbner basis of $(y^n-1,xy+x+1)$ in the ring $\mathbb{Q}[x,y]$, it gets worse and worse as $n$ increases. Any bound would have to be in terms of the degrees of the original generators as well as their sparseness, and I suspect that the overall picture is bad.
Overall your questions play to the weaknesses of Gröbner bases. You would need new ideas to make not just the computations of the bases, but also the actual answer numerically stable. You would also need new ideas to make Gröbner bases sparse.
You are probably better off with three standard ideas from numerical analysis: Divide and conquer, chasing zeroes with an ODE, and Newton's method. If you have the generators for the variety in an explicit polynomial form, then you are actually much better off than many uses for these methods that involve messy transcendental functions. Because you can use standard analysis bounds, specifically bounding the norms of derivatives, to rigorously establish a scale to switch between divide-and-conquer and Newton's method, for instance. Moreover you can subdivide space adaptively; the derivative norms might let you stop much faster when you are far away from the variety.
To explain what I mean by derivative bounds, imagine for simplicity finding a zero of one polynomial in the unit interval $[0,1]$. If the polynomial is $100-x-x^5$, then a simple derivative bound shows that it has no zeroes. If the polynomial is $40-100x+x^5$, then a simple derivative bound shows that it has a unique zero and Newton's method must converge everywhere within the interval. If the polynomial is something much more complicated like $1-x+x^5$, then you can subdivide the interval and eventually the derivative bounds become true. Also, with polynomials, you can make bounds to know that there are no zeroes in an infinite interval under suitable conditions.
You can do something similar in higher dimensions. You can divide space into rectangular boxes, and just median subdivisions. It's not very elegant, but it works well enough in low dimensions. In high dimensions, the whole problem can be intractable; you need to say something about why you think that the solution locus is well-behaved to know what algorithm is suitable.
Best Answer
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".