"Everything" is the intersection of all types (rather than the union, which is "Anything").

It does not means that it contains the values that all types contain. Rather, it contains the values that belong to every type. In other words, in order to belong to the type "Everything," a value has to simultaneously belong to every other type.

One way to think about this is that the name applies to each member of the type, not to the type as a whole. "Anything" is an accurate description of any possible value, so the set contains all values. "Everything" does not apply to the entire set, but to each individual element. An element must be everything all at once in order for us to describe it as "Everything." Nothing does that, so "Nothing" does that.

Some type systems also have a bottom type, and this has nothing to do with the bottom you're talking about.

If you consider subtyping, Bottom is a type IIRC, In the category with types as objects and subtyping relationship (a->b if b is subtype of a) as morphisms, Any is the initial object and Bottom is the final object of the category

And what is the type of that value?