Dragon Breath & Familiars

1) Does Dragon Breath last the full 1 min (10 rounds)?

2) Does Dragon Breath get around the "Familiars can't attack" clause by being an Area of Effect cone requiring a dex save?

3) If you use twin spell when casting on yourself and the familiar can both use their action to exhale the cone effectively doubling the spell damage? 

(1) Yes. As long as concentration is maintained.

(2) Yes. "Attack" is a specific action. Using a granted action is it's own action.

(3) Dragon's Breath can not be Twinned, per RAW/SageAdvice, but there are many GMs who ignore that ruling.