Methods are crucial for modularizing code and enhancing its readability and reusability. Discuss the significance of user$-$defined methods in Java programming. How do user$-$defined methods contribute to code organization and abstraction$?$ Elaborate on the role of parameters in methods, including both primitive data types and array parameters. How do method signatures impact method overloading and polymorphism, and how can these concepts be leveraged to design robust and flexible Java applications?