abortion and halacha
To summarize, Jewish law does not hold (as much of Christianity appears to) tha life begins at conception; rather, human life begins at birth. (Specifically at crowning, as I recall.) The torah covers causing the death of a fetus; it's a property crime. Killing a person, of course, is not. So abortion is permitted under Jewish law. Not desirable, but permitted.
Now here's where the halachic problem with the agenda of the far right comes in: under Jewish law, there are cases where abortion is mandatory. It is unambiguous that this is required to save the life of the mother; the rabbi generally agree that it is also required to preserve the health of the mother.
Most pro-choice folks (certainly myself included) argue on the grounds of individual liberty, but the author of this article points out that as Jews we should be considering the halachic issue, too. The rest of this entry is me talking, not the author.
If Bush gets his way (through legislation or by stacking the Supreme Court), we could end up in a situation where national law forbids the correct practice of our religion, specifically as regards to how we treat other people. I've been trying to think of an analogy for Christians (staying away from murder because it's emotional), and the closest I can come up with is: suppose the government required you to bear false witness against your neighbor, with the result that he would be criminally or economically ruined. (And if you think that can't happen...) That's a violation of one of the ten commandments. Would you be outraged? Would you heed that law? Requiring the Jewish community to stand by while an actual life is ruined in favor of a fetus is kind of like that. A Jew who does that violates laws (both between man and man, and betweeen man and God) that we take every bit as seriously as Christians take their laws that say life begins at conception. But Christians do not sin if they fail to prevent an abortion; we do in some cases if we fail to perform one.
If religion has no bearing on government, then the anti-abortion lobby has to rework its arguments. If religion does have bearing on government, then all religions must be considered, not just the one most popular with lawmakers. Some of the founders of the country may have been Christian (many were Deists), but this is not a Christian nation. Not then and certainly not now.

no subject
But am I correct in my perception that Christianity (limit that however you like) would never consider abortion to be mandatory?
That is true as stated because, even with double effect, direct abortion is not considered a moral act. Christians are called to treat the illness and failing to do that could be an immoral act.
If a method of extracting the fetus alive existed...
I'm sure that ethicists would be spending a long time working on the implications of that. My off the cuff response would be that it certainly seems to be an improvement over disposing of the fetus but I couldn't give a response more than that. Looking at my nephew and his dicey situation from being born at 27 weeks and the health consequences that he still carries 3 years later (and fortunately his health consequences are diminishing over time), my guess is that the technology to do this practically won't be available for a while yet. That is, of course, a guess. Until that time it would seem that the death of the fetus is attached to the act of abortion and conversation regarding abortion will most likely treat it as such. Even when one talks about double effect, there is a proportionality to the response and it requires a grave situation to get to the level where the death of the fetus can be a legitimate side effect.
I am unable to draw the specific lines in the questions that you raise but they are good questions. I do know that there are medical ethicists that study this sort of thing and provide guidance. If I come across a reference I'll send it to you. My short answer is that since double effect is involved, it must be messy. :)