Here is an idea to solve the problem, given 2 sets:
You can hold "sets" by a red black tree. In addition, for every node in the tree we associate one bit to determine if its subtree contains an element in both sets. For sake of presentation, it is called the insertion bit. I assume the red black tree sorts the elements from left to right.
When inserting an element to the tree, the algorithm checks if the element exists in the tree (i.e., in the other the set). If not- we insert the element as usual. If not, by traveling from the root to the leaf containing the element, the algorithm turns on the insertion bit of the corresponding nodes. In the worst case it takes $O(\log n)$.
When deleting an element, the algorithm checks if the element exists in the tree, and if the insertion bit is turned on. If the element does not exists in the tree- we return an error. If the element exists, and the insertion bit is off, then we delete the element as in the Red Black tree algorithm. Otherwise, by traveling from the root to the leaf containing the element, the algorithm turns off the insertion bit of the corresponding nodes. Deletion takes $O(\log n)$.
Finally, the algorithm for finding minimal element shared by both sets begins with the root. If the insertion bit of the root is turn off- then the sets are disjointed, the the algorithm returns an error. Otherwise, the algorithm travels recursively to the left child if its insertion bit is turned on, and otherwise it travels to the right child. The algorithm stops at the element with the minimal value. The algorithm runs at $O(\log n)$.
I am trying to think how to generalized the for a larger number of sets...