There are actually language mechanisms that can get you both. multimethods, for example, or their ultimate generalization, predicate dispatch. Few languages support them, though, because they are overkill for most application domains (though there are some, such as computer algebra, where they can be very useful).
Note also that while many OO languages constrain you in having methods only occur within a class definition, this is not a requirement. Multimethods invariably need to be more flexible (because there's no privileged "self" object), but the multimethod approach can also be applied to single dispatch. See also partial classes and related techniques.
Also, ADTs in their basic form are still a very specialized, limited form of sum types. Even if you want sum types, better extensibility is often needed (though ADTs remain a very useful special case), as well as the ability to actually describe subtypes of a sum type.
See how OCaml ended up with having polymorphic variants and extensible variant types and Scala with case classes, for example.
2. Multiple dispatch and static typing are completely orthogonal features.
3. Strictly speaking, you don't even need multiple dispatch. The key feature of multimethods that you need is the ability to add a method outside the class body.