Part 1, it depends on how good at keeping a secret Padame is, and how well she fains interest in other men, etc. In theory, all that the other senators know is that she was on her way to single motherhood. The whole "secret marriage thing" suggests that the Jedi are a chaste monastic society, so having a Jedi friend (which it seems many high ranking politicians on Coruscant did) necessary leads to the assumption that he's the father.
Even if you assume Bail does know who the babydaddy is, he only knows him as Anakin Skywalker, who Obi-wan killed on Mustafar. There is no particular reason to think Vader is a resurrected Anakin - him being a third secret apprentice of Palpatine is far more likely.