Yes. Well, most often the surgery does not have to be done that early. However, it can be done especially if the infant is not doing well. Way back in 1952, kirkland reported a successful operation in a 10 week old infant with coarctation and heart failure. http://circ.ahajournals.org/content/6/3/411.full.pdf.
Yes but. Generally, coarctation diagnosed in infancy is not mild and often requires surgery. It would be unusual for an infant to be diagnosed with coarctation in whom doctors thought a surgery would be necessary but that it could be "put-off" (i.e. Elective). Coarctation surgery is generally very safe and immediately effective, though there is a 10-20% recurrence risk (regardless of age).
Depends on history. It depends on severity of narrowing and other issues regarding the health of the baby. If not bad enough to cause compromise of growth, then delaying for as long as possible is desirable. This makes the surgery easier, is better for the patient, and will delay narrowing from scarring, or relative narrowing due to outgrowing the graft/repair.