Merging two splay trees whose ranges may overlap in $O(\log N)$
No, you cannot merge two splay trees whose ranges may overlap in  $O(\log N)$, where $N$ is the total number of nodes. You may have to move at least $\Theta(N)$ nodes in general or average situations.

It contains no information at all when we know "whose ranges may overlap". The ranges of any two splay trees may overlap.

It contains no information at all when we know all elements of A are not necessarily smaller than all elements of B. That is the same as all elements of A are not necessarily larger than all elements of B.