In this case the enemy is the code-maker and we are the code-breakers. As the code-breakers, our objectives are first to break the code and read the messages, and secondly to conceal the fact that we've broken the code, so that we can continue to read the messages, and also so that the enemy cannot take countermeasures. (Technical improvements in cryptography are reduced in effectiveness by technical improvements in code-breaking; the game remains the same.)
And so goes the give-and-take of intelligence gathering - it's all about who knows what. If we know what our enemy is saying among themselves, that's good. If the enemy finds out that we know, that's bad, because this leads the enemy to communicate truly important information by more secure means, and also this opens up opportunities for disinformation - if our enemy knows that we can snoop on one of their methods of communication, but we don't know that they know that, then they can use that method as a way of passing false information to us and having us believe it, damaging our ability to sort truth from fiction and thus to make use of intelligence in general. A civil trial with discovery in Padilla's case will reveal to the enemy some of our methods and means of spying on them - the result of which can only be positive for the enemy and negative for us.
With regards to your last question, I believe Padilla is accused of plotting to detonate a radiological weapon in the city of Chicago.
In the current situation, the code comprises all of the methods of enemy communication, not just technological ones.