Usually. I allow a pregnant woman to fly until 36 weeks. However if she is having any pregnancy complications such as high blood pressure or premature labor or the baby isn't growing well I strongly recommend against flying. If a woman flies late pregnancy wear support hose or move legs frequently to prevent leg swelling and risk of blood clots.
Yes. Most airlines allow pregnant women to fly up to about 36 weeks gestation. It is not considered unsafe to fly during pregnancy.
Yes. For most women with normal, healthy pregnancies, flying in the third trimester is just fine. We generally tell women not to fly after 36 weeks. After all, you could go into labor at any time, including on the plane. However, some problems during pregnancy make travel more risky. This is especially true for women who are at risk for preterm labor or have problems with the placenta.