I'm looking but not finding. Where are the rules for how different weapon qualities are activated?

Most of the qualities require 2 Advantage to activate, unless passive. The others are described in the Quality section.

2 advantage to activate weapon qualities. Qualities listed as "Passive" are always in affect.

The GM screen has a great chart with all the qualities and how they are used.

and spending advantage to activate weapon qualities doesn't affect the damage, correct? They get the damage boosts from advantage, and can use them for quality activation?

and spending advantage to activate weapon qualities doesn't affect the damage, correct? They get the damage boosts from advantage, and can use them for quality activation?

Advantage doesnt affect damage, unless you are using it to activate some sort of ability or crit

and spending advantage to activate weapon qualities doesn't affect the damage, correct? They get the damage boosts from advantage, and can use them for quality activation?

Damage is affected by uncancelled successes , so you'd be correct in that spending Advantage to trigger weapon qualities doesn't impact the damage of the initial attack. Autofire and Linked are triggered by Advantage, adding more damage by virtue of inflicting additional hits on the target.

Ah, thank you. Still getting a grasp on the dice in combat...

Also, each (uncancelled) advantage are spent on only one thing and then gone. So, if you spend 4 advantage activating a crit, you'd need 2 more advantage in that roll to activate a quality, or give boost die, or whatever.