Do tie fighters get to roll green dice against?

The bonus damage from Assault Missiles is direct damage, not an attack roll. Therefore, no, the TIE Fighters do not get to roll green dice against that extra damage if the initial missile attack hits.

Only the target gets to roll for defence. Every other ship (enemy and friendly) suffers an automatic 1 damage if the target is hit.

Only the target gets to roll for defence. Every other ship (enemy and friendly) suffers an automatic 1 damage.

IF the attack hits the target, after its defense roll.

A fully complete answer:

Lets say that the ship with Assault Missiles is Rhymer, for reasons, and the defender is Howlrunner, for others.

Rhymer attacks Howlrunner, and that goes thusly:

Rhymer spends his target-lock to fire the Assault Missiles, choosing Howlrunner as his target.

Rhymer rolls 4 attack dice.

If Howlrunner had any abilities or tokens to modify the Attacker's dice while defending, they happen here.

If Rhymer had any abilities or tokens to modify his dice while Attacking, they happen here.

Howlrunner rolls 3 agility dice.

If Rhymer had any abilities or tokens to modify the Defender's dice while attacking, they happen here.

If Howlrunner had any abilities or tokens to modify her dice while Defending, they happen here.

Dice cancelation happens

Evasions cancel [boom]s
Any remaining Evasions cancel [KABLAM]s

If any [boom]s or [KABLAM]s persist at this point, or if the attacker was Lt. Blount, the defender is HIT.

Damage happens

Shields cancel [boom]s

Shields cancel [KABLAM]s

If no shields, deal 1 face-down damage card per [boom]

If no shields, deal 1 face-up damage card per [KABLAM]

With Assault Missiles, if the attack HIT, each ship at Range 1 of the defender is dealt 1 damage. This is not an additional attack, and so is unable to be mitigated.