Secular reasons for and against legalising abortion

[Neo-Scholastic]

Going backward from birth, at what point was the entity in question not a human being?

Some questions should the fetus ever be considered a human being.
People are not required to donate organs, provide blood donations, etc. for other human beings even when those human beings are their children or close relatives. Should a pregnant woman ever be required to do so for an unborn human being by continuing the pregnancy? If so why?

This does not apply, since no donation actually occurs, in the sense that the woman is not being unnaturally deprived of any functioning organ. What the unborn receives from the woman is general sustenance in the form of food and protection from the elements. As a society we consider it the responsibility of caregivers to provide food, water and shelter for their dependents. To do otherwise would be neglect.

…is there ever a circumstance where continued pregnancy should be required even though it:
a) Endangers the life of the woman;
b) Endangers the long term physical health of the woman;
c) Endangers the long term mental health of the woman?

In instance a) there a trade of one life for the sake of another. That is a complex moral question that involves too many considerations and occurs under uniquely personal circumstances. No hard and fast rules apply. Reasonalbe people disagree and that is why very very few people would outlaw all instances of abortion.
Instance b) trades the life of one human being for the incapacitation of the other. Being dead is generally much worse that living with a minor disability. Rationally, I am inclined to speak in favor of continuing with the pregnancy; however, I think this is one on which reasonable and compassionate people can disagree.
Instance c) trades one human life for the emotional well-being of another. Given that depression and other mental illnesses are treatable conditions and highly variable, that would not be a fair trade.

…[if] the fetus has developed to the point that it could survive on artificial life support outside the uterus, must the fetus be kept alive outside the uterus, if the pregnancy is terminated? And if so, who is required to make arrangements and pay for such life support?

Yes, people are responsible for their actions and must live with the consequences of their choices. The couple responsible for the creation of the new life is financially responsible for it. I also believe in a social safety net that would provide assistance in the case of poverty etc.

If at anytime during the pregnancy a fetus becomes a human being, should child abuse and neglect laws apply before birth?

Yes. That seems appropriate although I think the specifics of the law matter.

Do any of the above answers change if the fetus can be shown to be severally physically or mentally handicapped, or that the fetus will have a very short life span after birth?

I think in instances a) and b) the relative health of the newborn would be a factor in making those kinds of tough judgments.