The +1000 ATK lasts as long as "Raging Flame Sprite" remains face-up on the field. "Raging Flame Sprite" can continue to get +1000 ATK each time it successfully attacks directly (becoming ATK 1100, then ATK 2100, etc.).
The effect of "Raging Flame Sprite" is a Continuous Effect that cannot be chained to.
The effect of "Raging Flame Sprite" is resolved after damage calculation and before Flip Effects are resolved (so the timing is the same as "Don Zaloog" and "Airknight Parshath").
If "Raging Flame Sprite" is equipped with "Twin Swords of Flashing Light – Tryce", its ATK will increase by 1000 points after each successful attack ("Raging Flame Sprite" must still deal at least 1 point of damage with each attack for its effect to apply). So the second attack will be with +1000 ATK, and it will gain another +1000 ATK after the completion of the second attack.