It could be interpreted that the NSA exists to catch criminals. Spying on the US government is a crime in the eyes of the US government, no matter where you conduct said spying. Counter-intelligence is an effort to catch those that spy on the US government, despite the fact that is likely illegal in the country of the person being spied upon. Although, what they do in terms of counter-intelligence doesn't always turn into criminal proceedings.
I think that the role of counter-intelligence is most of the time not really catch anyone, but to 1) get information from an adversary, and 2) develop methods which allow to protect own information against adversaries.
But I've always understood a part of counter-intelligence is to counter the intelligence gathering of an adversary, or I suppose ally as well depending on circumstances. Which is what you are saying with your second point. Therefore, there is someone to catch doing it, just in a broad definition of the term.
Spying on the U.S. is not a crime, and doesn't need to be. Between sovereign entities there is no rule of law, and sovereign entities have an inherent right to do whatever they need to protect their interests. It's a very different situation from the domestic criminal justice system.
As I said earlier, spying on the US government is a crime in the eyes of the US government. I think many a caught spy that was convicted for espionage would disagree with your statement that it is not a crime.
Yeah I'm having trouble with that argument too. How do you reconcile "countries interact in the lawless state of nature" with high-profile espionage prosecutions of non-citizens?