Engine failure on take-off is critical. From the performance point of view, the most critical time is at V1, but that is not usually difficult as a handling exercise (mine were all in the sim, happily my count in the real thing is a nice fat zero). At rotate, as you suspect, it can be more difficult, simply because tha aircraft is no longer restrained by the ground to stay wings level. With any twin, especially with swept wings which exaggerate it, the initial yaw will be followed by roll into the dead engine unless you do something. The best "something" is enough rudder to stop the yaw and enough aileron to return to wings level.
What you then aim to do is climb out at V2 and whatever rate of climb that gives you. There is no need to increase power; the performance calculations for any derated take-off account for obstacle clearance at the reduced power. The standard teaching is fly, navigate, communicate as the order of priorities. All companies have SOPs that cover these things, and the course of action in the event of engine failure is covered in the pre-take-off briefing; that will not cover the above handling points, but things such as flap retraction altitude, any route change as a result of losing the engine (often called emergency turn) and intentions as to where to land, which might be the departure field or might be another depending on weather, runway length, and other factors.
All my airline flying was Boeing, and yes, all that phase was always hand flown. Airbus 320 & later is a bit different, as the flight control system does some things that on other types are more like autopilot functions - but you will have to ask someone with relevant experience about that.