Is a woman who accidentally dropped a decorative bird's cage from her hand, resulting in its death, held accountable for this, and is she considered sinful for it?

There is no sin in the death of a bird without the intent to kill, for intent is a condition for accountability, and a person is not held accountable for error; as Allah Almighty says: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts intended. And ever is Allah Forgiving and Merciful."