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 weakness 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 need new ideas to obtain a sparse answer.
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.