The answer to the first question is positive, and the lower bound is achieved, since the set of all probability measures on $[0,1]^k$ with uniform marginals is compact and since the integral of the bounded continuous function $(x_1,\ldots,x_k) \mapsto x_1\cdots x_k$ on $[0,1]^d$ depends continuously of the probability measure.
Yet, finding the minimum is not obvious. For $k=3$, I intuitively expect that taking $X_2=1-X_1$ and $X_3$ equal to a decreasing function of $X_1(1-X_1)$ will give the minumum.