You can, but you may start to run into some limitations. Many languages have some version of type inference, which allows them to figure out the type of a thing, even though the programmer didn’t specify it.

C# has a very weak version of this with the auto keyword. Languages like Crystal take it much further by tracing the flow of data through the entire program. It generally works quite well, though there are a few edgecases that require explicit type annotations.

As for auto-completion, some languages feature designs that make it easy to offer auto completion even without type information. For instance, Elixir doesn’t have methods. You only have functions defined on modules, and it’s trivially easy to know what functions are defined on a module.

So it’s possible, but there are some limitations.

Yes! If your language allows you to put constraints on those types you can specify what specific types have access to. For example in Rust:

    fn foo<T: Positioned>(x: &T) -> Point {
       x.position()
    }

Note that this doesn't work in languages that use templates for generics, like C++, where templates work more like compile-time duck typing.

In dynamic languages functions aren't typically overloaded by argument type.

For OO languages and methods it does get a little guesswork-y, and tools often offer a wider range of guesses than is correct for completion. But you can show the class along with the offered completion, so it's not too bad.