As for visualization, I have seen SPD 2x2 matrices represented as ellipsoids (such as this one, which would be a 3D case), and the segment joining $A$ and $B$ in that Riemannian distance as a sequence of small ellipsoid slowly morphing one into the other (moving either in space or in time). It is a nice geometric visualization, but from a mathematical point of view it does not add too much insight.

I do not know of simple relations between the eigenvalues of $A$, $B$ and $A \#B$, apart from trivial cases where $A$ and $B$ have a common eigenvector.

This might be because in some sense the "natural" operation on that matrix space is not orthogonal conjugation $A\mapsto QAQ^T$ but generic conjugation $A\mapsto MAM^T$, with $M$ not necessarily orthogonal. As you can read in Robert Bryant's answer, a suitable transformation of that kind can diagonalize two generic matrices $A$ and $B$ at the same time. So "what is preserved under orthogonal conjugation" might not be the best question to ask, in an abstract setting. In a concrete setting where you have the matrices and you want to do computations with them, well, I understand perfectly the logic of your question but I do not have a ready answer for it.

(Incidentally, if you are doing computations with matrix means, you might be interested in Bruno Iannazzo's Matlab toolbox

As for visualization, I have seen SPD 2x2 matrices represented as ellipsoids (such as this one, which would be a 3D case), and the segment joining $A$ and $B$ in that Riemannian distance as a sequence of small ellipsoid slowly morphing one into the other (moving either in space or in time). It is a nice geometric visualization, but from a mathematical point of view it does not add too much insight.

I do not know of simple relations between the eigenvalues of $A$, $B$ and $A \#B$, apart from trivial cases where $A$ and $B$ have a common eigenvector.

This might be because in some sense the "natural" operation on that matrix space is not orthogonal conjugation $A\mapsto QAQ^T$ but generic conjugation $A\mapsto MAM^T$, with $M$ not necessarily orthogonal. As you can read in Robert Bryant's answer, a suitable transformation of that kind can diagonalize two generic matrices $A$ and $B$ at the same time. So "what is preserved under orthogonal conjugation" might not be the best question to ask, in an abstract setting. In a concrete setting where you have the matrices and you want to do computations with them, well, I understand perfectly the logic of your question but I do not have a ready answer for it.

(Incidentally, if you are doing computations with matrix means, you might be interested in Bruno Iannazzo's Matlab toolbox