The representation on Grey's Anatomy is completely false all the way around. Except the fact that peds surgery is a rotation for the surgery residents and it requires a fellowship after general surgery. But they get half the other surgical subspecialties wrong, so...
The peds surgeons will do some management of surgical patients... how much will vary by institution. At some, the pediatricians manage them on the floor (one peds program I looked at had automatic peds consults on every child under the age of 12 on a surgical service); at others, they have NPs or PAs doing the floor work, and at still others, they'll do it themselves. They will sometimes manage kids with multiple medical problems that have been surgical patients in the past or have a problem that can be managed with surgery (but will try medical management first), and may manage trauma patients, depending on the severity of the trauma (and the age of the patient). Generally, though, the medical management of the patient falls to the pediatrics team.
Pediatric intensivists, as mentioned above, can do just about everything up until and after the surgery. They can intubate, put in lines and tubes, etc. General pediatricians generally don't do as many procedures.