I think I may have missed something, but the damage of the Ancient Sword of +2, is that added to Brawn, like a Melee weapon?

Or if another attribute is being used for Lightsaber checks, like Intellect for Soresu Defender, would the +2 Damage be atop my Intellect score?

+2 is added to Brawn, as it is a melee weapon that uses the Lightsaber skill.

The various lightsaber talents that allows Lightsaber skill checks to be made with other characteristics than Brawn only applies, as I understand it, to the combat check, not the weapon damage before successes from the check are applied. I may be wrong, but I don't think so.

Above is correct.

Yup, Jegergryte is correct. The only difference between the Ancient Sword and other Melee weapons is that the Ancient Sword uses the Lightsaber skill when making combat checks. Damage is calculated normally as for all Melee weapons.

Which means your ataru striker with a high dex who gets a lot of successes will do brawn+2+number of successes on a agility_lightsaber skill roll.

Which means your ataru striker with a high dex who gets a lot of successes will do brawn+2+number of successes on a agility_lightsaber skill roll.

Of course, the Characteristic used for the dice pool can variably affect the damage output in such a way. So yes, you'll be hitting harder, or perhaps "more accurately," if you use you optimized Technique talent.

Still, the damage rating of the weapon is simply Brawn + 2.