I have not worked at the federal level for many years so I may need to be corrected, but at the state level you would be reading far too much into this response. The automated email probably meant that it itself could not receive replies; that it was automated and did not have the capability to receive messages back. You were perfectly correct in sending the thank you notes.
Asking for feedback on your interview performance was fine. It is also true that you were asking those people for a favor. They are trying to fill a position, and are probably looking for good candidates for future vacancies as well, but they are not your employment counselors. If they do not have time to explain to every applicant why the one they chose better fit their needs, I would not take it personally.
If you want to work for that Department, you should certainly apply for future vacancies as well as make a point of learning as much as you can about how it is organized and what it does. If you have contacts within the Department I would cultivate them, but I would not keep trying for feedback on what may have been an impersonal decision.
In this market we have many well-qualified and talented people at the state level for each opening and I am sure this is true at the federal level as well. If each person who is not selected takes it as a personal affront or a failure on their part, they are doing themselves a great injustice. I would keep applying for positions you want and try not to take setbacks as anything but challenges to improve your presentation.
It is good to ask for feedback and show an interest in future positions, but to insist that busy people help you get a job can soon become an imposition, unless they are employment counselors or personal friends.
As I say, I have no recent experience at the federal level so better informed respondents may have something else to say. Good luck in your future endeavors!