The algorithm is open ... however it was produced by a politically motivated government branch that seems to harbouring a cynical bunch of criminal bastards ... do your own due diligence, if you don't have to deal with them why bother?
Well we do have to "deal with SHA-2" as a change to a different hashing algorithm would be a hard fork and that isn't going to happen.
Actually I believe it could happen. And it should happen if SHA-2 was compromised, that was always the plan but right now
SHA-2 is NOT compromised. The NSA may have some shortcuts and could exploit random number generator issues (including backdoors in those) but a clear get the private key from the public one is not possible at this time.
Changing the algo was ALWAYS planned in Bitcoin if it was compromised.