trait Animal {
    fn new(name: String) -> Self; // Self is the type that implements the trait
    
    fn name(&self) -> String;
    
    fn noise(&self) -> String;
    
    fn talk(&self) {
        println!("{} says {}", self.name(), self.noise()); // This is default impl.
    }
    
    fn specie() -> String;
}

impl Animal for Dog {
    fn new(name: String) -> Dog {
        Dog { name: name }
    }

    fn name(&self) -> String {
        self.name.clone()
    }

    fn noise(&self) -> String {
        "bark!".to_string()
    }
    
    fn specie() -> String
    {
        "Canine".to_string()
    }
}

impl Animal for Sheep {
    fn new(name: String) -> Sheep {
        Sheep { name: name, wool: true }
    }

    fn name(&self) -> String {
        self.name.clone()
    }

    fn noise(&self) -> String {
        if self.wool {
            "baaaaah!".to_string()
        } else {
            "baaaaah?".to_string()
        }
    }
    
    fn talk(&self) {
        println!("{} pauses briefly... {}", self.name, self.noise());
    }
    
    fn specie() -> String
    {
        "Ovine".to_string()
    }
}

fn test<T: Animal>(animal: &T) {
    println!("{}", T::specie());
    animal.talk();
    let clone = T::new("Clone of ".to_owned() + &animal.name());
    clone.talk();
}

fn main() {
    let my_dog: Dog = Animal::new("Snuppy".to_string());
    let mut my_sheep: Sheep = Animal::new("Dolly".to_string());
    test(&my_dog);
    test(&my_sheep);
}

println!("{}", T::specie());

trait Animal {
    fn new(name: String) -> Self; // Self is the type that implements the trait
    
    fn name(&self) -> String;
    
    fn noise(&self) -> String;
    
    fn talk(&self) {
        println!("{} says {}", self.name(), self.noise()); // This is default impl.
    }
    
    fn species() -> String;
}

impl Animal for Dog {
    fn new(name: String) -> Dog {
        Dog { name: name }
    }

    fn name(&self) -> String {
        self.name.clone()
    }

    fn noise(&self) -> String {
        "bark!".to_string()
    }
    
    fn species() -> String
    {
        "Canine".to_string()
    }
}

impl Animal for Sheep {
    fn new(name: String) -> Sheep {
        Sheep { name: name, wool: true }
    }

    fn name(&self) -> String {
        self.name.clone()
    }

    fn noise(&self) -> String {
        if self.wool {
            "baaaaah!".to_string()
        } else {
            "baaaaah?".to_string()
        }
    }
    
    fn talk(&self) {
        println!("{} pauses briefly... {}", self.name, self.noise());
    }
    
    fn species() -> String
    {
        "Ovine".to_string()
    }
}

fn test<T: Animal>(animal: &T) {
    println!("{}", T::species());
    animal.talk();
    let clone = T::new("Clone of ".to_owned() + &animal.name());
    clone.talk();
}

fn main() {
    let my_dog: Dog = Animal::new("Snuppy".to_string());
    let mut my_sheep: Sheep = Animal::new("Dolly".to_string());
    test(&my_dog);
    test(&my_sheep);
}

println!("{}", T::species());

For example, I have 2 classes, J and K, both of which implement interface I and in some circumstances may be used in the same place.

if the type itself can be passed I could have a function that takes a type implementing I and call whichever version of a static member function, based on the passed object

This, of course, has the drawback of forcing you to enumerate the possible types.

Another thing that gets close is static interface methods. Java has interfaces with static methods, but you can't override those. So we need a language with has something like interfaces with static members that we can override. Rust is such language.

One thing I haven't seen (…) is using a type as a value that can be passed around, allowing it to instantiate new objects, call static functions etc, while still providing all the benefits of strong type checking

Emphasis mine.

For example, I have 2 classes, J and K, both of which implement interface I and in some circumstances may be used in the same place.

if the type itself can be passed I could have a function that takes a type implementing I and call whichever version of a static member function, based on the passed object

Emphasis mine.

This provides "all the benefits of strong type checking". However, of course, has the drawback of forcing you to enumerate the possible types.

If the type is to be passed at runtime - for example, in a variable, as the question suggests - it means it is not known at compile time, so we lose that strong type checking. To pass a type at runtime but have some type information in compile time, we have generic type arguments and constraints, of course. Yet, that won't give you access to static members.

I only see two paths to keep such strong type checking: We interrogate the object, as shown above. Or stronger generic constraints…

Another thing that gets close is static interface methods. If we can make a generic constraint to such interface, we could be able to use those static members.

Java has interfaces with static methods, but you can't override those. So we need a language with has something like interfaces with static members that we can override. Rust is such language.