All five of your examples are grammatically correct. (1) and (3) are ambiguous. In both of them, "as" can mean either "because" or "in the way that".
(4) is awkward because of its redundancy, which makes (3) awkward as well when the "because" meaning is applied to "as". It sounds better simplified:
4a: Nobody else could have saved them, so I did.
4b: Nobody else could save them, so I did.