st.statistics - Expected distance between two points with missing coordinates

You can rephrase your question as follows: first we subtract the known vector from both and then take care of the known coordinates. So assuming the coordinates of the two points are $(alpha,beta)$ and $(gamma,X)$ where $alpha,gamma in mathbb{R}^m$ are known, and $beta in mathbb{R}^n$ is known, but $X$ represent the unknown coordinates constrainted to lie inside the cube $[-1,1]^n$, the integral to evaluate becomes
$$frac{1}{2^n}int_{[-1,1]^n} sqrt{ |alpha-gamma|^2 + |beta - X|^2 } dX$$
In the lower dimensional case this can be integrated. But an analytical expression in higher dimensions is elusive. For the case $alpha = gamma$ and $beta = 0$, some bounds were obtained in an old paper of Anderssen et al. http://dx.doi.org/10.1137/0130003 For more general probability distributions there is a recent paper with some bounds by Burgstaller and Pillichshammer. http://journals.cambridge.org/action/displayAbstract?aid=6622208

Of course, one can get a fairly trivial bound by Cauchy-Schwartz
$$ int_{[-1,1]^n} f(X) dX leq 2^{n/2} left( int_{[-1,1]^n} f(X)^2 dX right)^{1/2} $$
and that
$$ int_{[-1,1]^n} R^2 + |beta - X|^2 dX = 2^n (R^2 + beta^2) + int_{[-1,1]^n} X^2 dX$$
the last term is simply evaluated as $n 2^n / 3$, so putting it all together we have the upper bound for the expected value by
$$ frac{1}{2^n}int_{[-1,1]^n} sqrt{ |alpha-gamma|^2 + |beta - X|^2 } dX leq sqrt{ |alpha -gamma|^2 + beta^2 + frac{n}{3}}$$
which is slight improvement over the utterly trivial upper/lower bound of $sqrt{|alpha-gamma|^2 + beta^2 pm n}$ if you just maximize/minimize each coordinates.