Rulings

Cryptolith Fragment’s second ability is a mana ability. Players can’t respond to it or to the loss of life it causes.
If any player has 11 or more life as your upkeep begins, Cryptolith Fragment’s last ability doesn’t trigger. If any player has 11 or more life as the ability resolves, the ability has no effect.
In a Two-Headed Giant game, Cryptolith Fragment’s mana ability causes each player to lose 1 life, so each team loses 2 life.
In a Two-Headed Giant game, Cryptolith Fragment’s last ability checks whether each team’s life total is 10 or less.
In a Two-Headed Giant game, Aurora of Emrakul’s last ability causes each opponent to lose 3 life, so the opposing team loses a total of 6 life.