What genetic disorders can be recognized in pregnancy?

Several. The most common ones are chromosomal problems like down's syndrome. The chances a woman will have a chromosomally abnormal baby go up with age. Whether you do the genetic tests or not depends on whether you want this information or not.
Depends, hundreds. This depends on how far you are willing to go to get the information. The best information is gained by obtaining fetal cells by sticking a needle into the womb & drawing them off for study. Chromosome studies and enzyme studies can be done. If you are looking for a specific problem the tests can be arranged in specialty labs. There is no one test that would catch them all.

Related Questions

Is it possible to detect any genetic disorders in the baby by ultrasound during pregnancy in second trimester?

U.S. vs blood test. The 2nd trimester ultrasound can pick up anatomical abnormalities, look for placement of the placenta, help date the baby, and confirm the number of babies in the uterus. The "quad screen" which is a blood test of the mother, can pick up common genetic abnormalities. Talk to your ob, midwife for more details. Read more...
Poor sensitivity. A high resolution ultrasound can often yield evidence of structural changes that occur more often in some chromosomal disorders.It is not specific.Many genetic metabolic disorders are only picked up after birth when byproducts of metabolism build up in babies blood.Some metabolic disorders can be confirmed by fibroblast culture on material from an amniocentesis when family hx is suggestive. Read more...