My theories for this season and comments about last night's episode.
The blood from the eyes is related to a Lizard. Hence the title of the episode "Lizards", I think it applies to more than Claire being able to regenerate. When horned lizards are scared they secrete a blood substance from their eyes that will repulse their predators. Its seems like Maya does the same thing. I don't think its a coincidence. Her twin brother is the only one to calm her. I think he absorbs her fear, which is how their friend was healed, but I do not think he is a healer like Linderman.
As for Peter not using his abilities to steal the box, it seems like he only uses his powers as an instinct rather than intentionally, which would be why he couldn't just take it. He doesn't remember how to control or use them.
I really want to know what Mama Petrelli's power is (assuming that she has one, since both of her children do). My guess is to know people's dreams or perhaps to plant ideas in someone's mind. Something in Season 1 gave me that impression, but I don't remember the exact situation.
What if "The Boogeyman" was 'created' by the older generation - Hiro's Father, Linderman, Mama Petrelli? When the company would kidnap people they could have taken some kind of blood sample and transfered it to an individual creating a super person, who then turns against them? It seems like it could be a comic book theme. What if that person was the child of Mama Petrelli and Hiro's father? That could have explained the surprise when he saw the person who pushed him over the edge.
Of course this is all speculation on my part.