Do doctors screen for metabolic disorders after birth?

The state does. In the US, each state screens different metabolic disorders... most will screen for the most common ones like PKU. Testing is done at the state lab and abnormal results are communicated to the primary doctor and in some states to the expert panel.
Varies. In the US each state has basic testing for metabolic disorders. The number varies by state. In my state samples from the hospital before discharge and 2 weeks are submitted for about 27 separate tests.
Family Hx most! For a\child, the family hx is most important. If the mother had pre- or full eclampsia, or gestational diabetes that is quite significant. As are the parent's hx for truncal obesity, cholesterol problems, diabetes, an mi's.
Which one! Astute doctors can give a heads up to a young infant by looking at the parents! Are they obese, hypertensive, have bad cholesterol. Did the mother have hypertension or gestational diabetes during pregnancy. Is there premature heart disease or adult onset dm. Metabolic syndrome and all complications are best predicted by these questions. Type 2dm is much more hereditable than type1.
Yes. Most states mandate newborn testing for a large panel of rare but serious metabolic disorders before the baby is sent home.