1) When Barbarians destroy a city, do you get a free Settler, too?
I don't think so, since it is only mentioned in the conquering city part of the rules.
Correct. They do not.
2) When you advance Steel Weapons using 2 Ideas (and also pay 2 Ore), does Free Education trigger?
I could see this go either way.
Yes, free education would trigger
3) When you get an advance for free (i.e. via Science or getting it in the Status Phase), does Free Education trigger?
No it would not. You are not paying for the advance