Nicholas,
Takeoff configuration means the way the airplane is set up with respect to flaps, leading edge devices, etc. When an engine fails, no configuration changes are made.
Where the takeoff is flown manually, one simply flies the airplane. No big changes, nothing different except some rudder input to offset the assymetric (uneven) thrust. The takeoff continues just as it did before, but things will happen a little slower. Reaching the takeoff safety speed of V2 will take a little longer. One's final climb pitch angle will be shallower than the pitch angle with both engines operating, and the climb rate will be less.
Normally with an engine out, nothing is done until at least 400'. At some airports, a turn procedure is prescribed, and other than a radio call to notify the tower or departure control that the crew is handling the problem (and to announce an emergency, where applicable), little else is done. The airplane will typically level off or reduce climb to begin cleaning up, and then continue climbing while flying either a departure procedure, vectors to return to land, or a designated engine-out procedure ("turn procedure"). During this time, once cleaned up (configured for the climb), the appropriate checklists will be run, abnormal or emergency checklists will be run, the engine secured or other problems addressed, and a preparation made to take the next step. In most cases, the next step will be a return for landing.
In that event, generally a check of weather will determine that we can indeed return, we'll need to dump fuel, we'll begin that process, brief the approach, get a clearance, and return to land.
The takeoff process with the engine out isn't much different than with the engine running. It's just a little (or a lot, depending on the airplane and the amount of thrust used) rudder, and straight, simple flying the airplane.
In our operation, we never engage the autopilot until our flaps are up (configured for the climb, or "cleaned up"), so the entire evolution is hand-flown.
Yes, engine-out on takeoff is the most critical time for an engine failure.
The engine-out doesn't need to be at rotation; it can occur at or any time after V1. The takeoff will still continue to the preplanned rotation speed, and the climb-out will be at V2.