Doppler ultrasound is an ultrasound that looks at blood flow. The hand held doppler detects the blood flowing through the heart, it is still an ultrasound but gives you sound rather than an image. A full size U/S machine will give you sound as well as images, ie. more information. A uterine doppler is looking at the blood flow in the placenta and umbilical cord making sure it is sufficient for bubs. And sometimes includes fetal doppler looking at blood vessels inside the baby. These, as you can imagine, can be more difficult to obtain as babies rarely lie still on their backs when asked.

During the morphology scan they will take all the bubs measurements, so will get the growth information then. It is normal to check the length of the cervix during a morph. scan but perhaps the dr wrote it on the request to ensure it was done. The uterine doppler could also be done at the same time, but perhaps the person who is most experienced in high risk u/s's isn't available to do it all at once? I'm not sure. But how long it takes will depend on the information they require, how easy it is to obtain and whether or not they are doing fetal doppler as well. But at a guess, i'd say about the same time as your morph scan, perhaps a little more or less!!!