(1) The paramount objective is to save lives, if this can be done without violating any exceptionless moral norms.
(1) Since the woman's life cannot be saved, that is off the table as a duty. (Something that is impossible cannot be obligatory.)
(2)The duty that remains is, saving the children.
(3)What will happen to the woman if you push her off the stairs? That is unknown. She could land someplace soft and be uninjured. She could be bruised. She could break a bone. She could break her neck and die. Any of these outcomes are foreseeable, but unintended. The one thing intended is to move her away from the top of the stairs.
(4) Will pushing her over the side of the steps, kill her because she'll be burned in the fire? No, because she will be burned in the fire in any case. At the top of the steps, she is doomed to die in the fire. At the bottom, she is still doomed to die in the fire. Pushing her off there steps is neutral with regard to fire fatality hazard.
(5) My conclusion. Push her off the steps.
If there is any possibility of doing so (in this seconds-count situation) try to minimize the harm of her fall. If there is a mattress handy, shove it over for her to land on. (But if you can't, you can't.)
After the children are out, go back and see if you can drag her out of the house on the mattress. (But if you can't, you can't.)
I see it as essentially a triage situation. Your "perfect" intent is to save everyone--- and you will, if things go "perfectly." But you know in all probability that won't happen. So you save the two children you can. Then (if remotely possible) you go back and try to save the woman. (But if you can't --- say the ceiling has collapsed --- you can't.)
What do you think of this?
There is a question of freedom to act. In the example, when a mother steals an apple to fit the child, I indeed did not clarify the presumption that for a mother there is no choice but to feed the child. If she has a choice: the child is crying but not in danger of death, then she may not grab that apple. Another example would be if the private property is not marked: then, perhaps (depending on the acculturation) an intruder is not acting freely simply because he does not know that the apple is private and cannot be taken. (The latter was the situation with the American Indians who simply could not grasp the concept of private land for agricultural use).
Back to your example: the firefighter is a professional, trained to evaluate the situations and consequences of them. So, he will be killing the fat woman freely by pushing her off the stairs. His duty is above all not to kill anyone; this duty remains even though she is about to die anyway. His duty to save the children is satisfied: he made an attempt and discovered that he could not do so without committing a murder. If the child pushed the woman, or a parent, then the moral equation would have been different because the child or parent would be acting in a highly agitated state and not freely.
Likewise, a doctor may attempt to save the life of the mother in labor, but he may not kill the baby in order to treat the mother: both the baby and the mother are his patients.
Likewise, a priest who is told at confession that the penitent is about to commit a heinous crime may not use the information the priest obtained to stop the crime.
These are all cases when double effect provides no excuse because a positive step is taken freely and in the knowledge that the step is immoral. The fact that the criminal step serves a greater good does not excuse the step.