default methods in interface but only static final fields

All fields in interfaces in Java are public static final.

Even after addition of default methods, it still does not make any sense to introduce mutable fields into the interfaces.

Default methods were added because of interface evolution reasons. You can add a new default method to the interface, but it only makes sense if the implementation uses already defined methods in the interface:

public interface DefaultMethods {

    public int getValue();

    public default int getValueIncremented() {
        if (UtilityMethod.helper()) { // never executed, just to demonstrate possibilities
            "string".charAt(0); // does nothing, just to show you can call instance methods
            return 0;
        }

        return 1 + getValue();
    }

    public static class UtilityMethod {

        public static boolean helper() {
            return false;
        }
    }
}

No - in Java 8 all fields are static and final as in previous Java versions.

Having state (fields) in an interface would raise issues, in particular with relation to the diamond problem.

See also this entry that clarifies the difference between behaviour and state inheritance.