I have the conjecture that the volume of the intersection between an $n$dim sphere (of radius $r$) and an ellipsoid (with one semiaxis larger than $r$) is maximized when the two are concentric, but still did not find a way to prove it. Any suggestion?
From a result of Zalgaller, this is true for any two centrally symmetric bodies. (Here is his lecture which inculdes this topic.) Namely, assume that the center of first body is at $0$. If $\vec r$ is the center of the secoind body and $v(\vec r)$ is the volume of intersection with the first one then according to Zalgaller's theorem $v>0$ in a convex domain and inside of this domain the function $v^{1/n}$ is concave. Clearly $\vec r=0$ is an extremal point of $v$. Thereofre $0$ is the point of maximum. 


By a rotation, you may assume that the axes of the ellipsoid are parallel to the coordinate axes. Choose the first coordinate of the center of the ellipsoid, and translate the ellipsoid so that this coordinate becomes zero, keeping the other center coordinates fixed (this corresponds to symmetrizing about this axis). The volume of the intersection increases when you do this, since the intersection of an interval of fixed length with an interval centered around the origin is maximal when the interval is centered about the origin too (this is the 1dimensional case of your question). Repeat with the other coordinates, until the ellipsoid is centered at the origin, and the volume of intersection is maximal. 

