It came across to me like they were trying to imply Winn did it because of her jealousy of the Emissary's faith in the prophets and his willingness to trust them with his son's life. Which doesn't quite make sense.
Winn being jealous the prophets chose Kira makes more sense, but I didn't see any suggestion of that motivation in the episode itself.
To your first point: that could be part of it. Winn always wanted to have the type of close connection to the Prophets that Sisko has.
To the second point: I seem to recall some suggestion of this in the episode, but it's been a while, so I can't recall precisely.