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?
How might they recover from this blunder?
One way would be to spread FUD about SHA-2 to convince everyone to switch to a new algorithm their deep cover agents had prepared just for this event.
But on the other hand, if SHA-2 was broken and they wanted to keep the truth from getting out, they'd propose a story just like what I wrote above. Unless that's just what they want you to think.
Maybe this loop of infinite recursion of motives but no proof is not the way to go.
Instead, look at this another way.
There is an enormous financial incentive to being able to break double SHA-256. The the most obvious incentive belongs to the ASIC manufacturers, who are devoting a lot of time to building machines that try to break double SHA-256 as rapidly and efficiently as possible.
None of them have found a substantial shortcut yet, despite years of working on it.
If the NSA did have a secret method, then every single person in the organization who knew about it would have a huge incentive to profit from it personally. Could all of them resist the temptation?
I think the hash rate will tell us if/when SHA-256 is broken, because we'll see a sudden increase that's not explainable any other way. Unless or until that happens, SHA-256 is probably safe.