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'.
That still doesn't prove that NSA has intentionally made SHA insecure. It gives them a motive, but there's no evidence.
Your fallacy is in where the burden of proof lies.
Is it better to trust the good intentions of the nsa, or to use a clean algorithm so there is no need to trust them?
Do they have such a sparkling history that it is wise to trust them?
OK, if you were in control of the hashing algorithm used by Bitcoin, which one would you use and why?