Sorry I guess I really only answered your last question there :) Let me address the others real fast as well.
gravenewworld said:
At what point during development does the embryo start producing blood?
Erythropoiesis, the technical name for red blood cell (RBC) production starts early on in the fetus, primarily in the yolk sac. Around 3-4 weeks. This function moves to the liver and spleen around week 7 and stays there for the most part to around birth. As you get closer the birth the bone marrow starts to play a larger roll, where it will occur in adults.
gravenewworld said:
The umbilical vein brings in oxygenated blood from the mother to the fetus' and also deoxygenated blood from the fetus to the placenta.
The mother's oxygenated blood doesn't actually come into contact with fetal blood (from a simplified view I guess, some fetal blood, particularly monocytes do escape into maternal circulation, but well let the histologists continue arguing about that) it fills small lacuna derived from invasive trophoblastic tissue (really neat tissue there by the way, its almost like controlled neoplastic cells!) and exchanges respiratory gasses and waste products across the placental membrane.
The umbilical vein attaches to the placenta and enters the body through the umbilical ring (soon to be your "belly button") and carries
fetal oxygenated blood (not maternal blood) to the fetal liver. In the liver it links up with deoxygenated blood from the IVC (inferior vena cava) and is returned to the fetal heart (right atrium). Since in the fetus the oxygen isn't coming from the lungs, the blood is shunted from the right atrium to the left atrium via the foramen ovale (consequently what most people refer to when they say that the babies "heart wasn't closed" at birth, which resolves itself 99% of the time) and then out to the rest of the fetus. The blood ends up in the umbilical arteries (branches of the internal iliacs) and then back to the placental where it gets reoxygenated.
At birth the crimping of the umbilicus leads to reduced pressure in the venus side of fetal circulation and right side of the heart. This pressure decreases below the pressure of the atrial side and left side of the heart, which closes the foramen ovale redirecting blood flow from the right atrium to the right ventricle and through the pulmonary vasculature where the newborn's lungs will take over oxygenation.