why Array.length has no () but String.length() does?
At the syntactic level, it really comes down to "historical reasons" and "because that's the Java (pre) 1.0 language designers decided to do it". The only people who can truly tell you why are the language designers themselves.
However:
The length field is special in the sense that it is actually in the array object's header, and fetching the length of an array uses a special bytecode.
The length field cannot be changed by any means short of dangerous native code hacks. It is really important that this be so, to avoid problems with heap corruption, etc. (By contrast even a final on a normal object can be modified using reflection.)
In these respects, length on an array is genuinely different from any other field.
Since the early days of (pre-)Java, the use of exposed fields in APIs has gone right out of style.
Most people would say that this is a good thing. By exposing the length information as length() and size() methods on strings, collections, etc, the class library design allows a Java implementor to change the representation of the respective types. That is not possible (in Java) if fields are exposed directly.
