1) Does the NSA have any interest in breaking bitcoin?
Of course.

2) Do they have the means? Do they have any influence over the cryptography?
Yes. Sha is their creation and they made special adaptations to it for reasons that are secret.

3) Has the NSA ever engaged in a similar type of deception, i.e., promoting weak cryptographics so they could seem to be breaking codes, doing their jobs, expertly?
They have. They are not so much 'code breakers' as 'con men employing code breakers who are willing to work for con men'.