Can I be held legally responsible for killing a dog that ran out in front of my car?

Whenever you have a question as to whether you are legally responsible for an accident resulting in an injury remember that the standard is whether you were negligent. Most states have a breakdown of the factors that must be proven before it can be said you were negligent. In general think in terms of [1] was there a duty owed (ex: drive in a manner to avoid accidents and obey the law) [2] was there a breath of that duty (drunk driving, texting while driving etc) [3] was there a harm as a result of the breath [injury, death etc) [4] did the harm result in damages (medical bills, damage to property, etc).  If all are yes then there is a good likely hood that you may be found to have been negligent i.e. legally responsible. The laws vary with respect to the state you live in but for the better part if [1] and [2] can be answered yes see an attorney right away.

In order to be responsible, your neighbor would  have to prove that you were somehow negligent.  If you were obeying the traffic laws then I dont see a view of the accident where you would be liable.

From your description of the accident, I think that she was responsible for her dog's death, if she allowed it to run free outside. You can politely tell her if she calls again that you are sorry her dog died, but that the accident was unavoidable as the dog ran quickly into the road. The fact is that the dog should not have been able to run into the road and should have been in a protected environment or on a leash when outside.

It is unlikely that you are responsible.  From your side of the story it does not sound like you did anything wrong or negligent.  Therefore, unless you live somewhere with some odd dog laws that makes anyone who hurts a dog through no fault of their own responsible for the harm to the dog, then you should be fine.  

However, as always, it is prudent to contact an attorney.