First, lets be absolutely clear that you said absolutely nothing about dealing with families of sets.
The link you provide is discussing union and intersections of families of sets.
It is easy to get different mathematicians into an argument over this question.
Herbert Enderton remarks that trying to define is like division by zero it is best to say it is undefined. But he adds that logicians particularly do not like that. That is what you see at ask DrMath. In his text, Stoll restricts intersections of collections to non-empty collections.
In so called naïve set theory, we have this rule: for all .
In that case, it follows that .
As to your second question: .