Do ultrasounds let doctors measure babies?

Yes. Ultrasound machines are invaluable in managing pregnancy today, especially high-risk pregnancies. Not only do we measure the baby's size (head, abdomen, thigh bone), but we can evaluate all of the internal organs, from brain to bladder. We also can make sure the house is in order -- evaluating the placenta and the amniotic fluid quality, even measuring the cervical length.
They can. Ultrasounds during pregnancy provide valuable information to the doc. Depending on the equipment & tech, you can locate the placenta which can pick up real problems for delivery. The head size & bone length can be used to validate the expected due date. Some can pick up kidney, heart, or other problems in the baby.