Can a hysterosalpingography open tubes?

In theory, but... Theoretically yes, hsg could flush tubes open. In practice this might happen more often than we're aware. But, hsg has not been shown to change pregnancy rates consistently, so it should be considered a test and not a treatment. A few small studies suggested possible small improvements in pregnancy following hsg, but these have not been confirmed by larger better studies.
Yes. More likely if use oil based contrast. See radiologyinfo.Org.