Claim #1 - "The leading cause of death for women worldwide is due to pregnancy and childbirth."
I find that plausible, though I admit to not knowing the facts.
The reason I find it plausible, is that as a grad student I walked through some of the cemeteries in New Haven, looking at the enscription on the tombstones. And what stood out was the large number of deaths of relatively young women, related to childbirth.
These were from before we had antibiotics available. I'm sure the situation is very different now. But they might be a good indication of what happens in parts of the world where medical care and antibiotics are not readily available to most of the population.
Wiki contains a chart ...
It is difficult to know how to read that chart. If a woman becomes infected as a result of childbirth, and dies of that infection, will it be counted in the chart as childbirth related, or will it be counted as death from an infectious disease?
Claim #2 - "we have done those studies where we have taken a random, representative sample of women who have had abortions and followed them for several years. And very, very few of these women suffer depression. Furthermore, the ones who do have problems are likely to have not been emotionally stable before they chose to have the procedure."
I distrust most of the data on this question. Both sides of the debate have axes to grind. Attitudes are sufficiently polarized, that it is difficult to know which surveys are honest. And, even then, the act of following a woman might itself affect the outcome.