Why do min/max operations in van Emde Boas trees run in $\mathcal{O}(1)$?
No. They are not given as an input! They were stored and updated as an attribute of each tree(Beware that the concept of the "attribute" in real-world programming means a "variable" in a programming language. More details depend on the programming language that you have chosen up to now to develop the data structure). The same situation is taking place in the min[Max]-heap data structure. You can find the minimum [maximum] in a min[max]-heap with $O(1)$ operation by keeping update (in insertion time) the mentioned attribute.