You activate “Cross Counter” before damage calculation but after flipping the attacked monster face-up. The “Cross Counter” card is sent to the Graveyard after resolving its effect and destroying the attacking monster.
The effect of “Cross Counter” only applies to the one attack and does not continue for the rest of the turn.
If your opponent’s “Amazoness Swords Woman” attacks your Defense Position “Mystical Elf”, and you activate “Cross Counter”, you (the controller of “Cross Counter” and “Mystical Elf”) take 1000 points of damage due to the effect of “Amazoness Swords Woman”, and then “Cross Counter” destroys “Amazoness Swords Woman”.
“Cross Counter” is not an effect that targets.
The effect of “Cross Counter” that destroys a monster does not start a chain.
If your “Stone Statue of the Aztecs” is defending and you activate “Cross Counter” the Battle Damage your opponent takes is only doubled once (by either the effect of “Cross Counter” or “Stone Statue of the Aztecs”).