Which speed has to be remained after getting airborne and is it controlled by the FMC or is it set by flight crew.
It's calculated by the FMC based on weight and flap setting and confirmed and selected by the flight crew.
I understand the speed setting after t/o has to be around V2 + 15 kts
Yes, you keep this speed until acceleration altitude (usually 1000 feet above airfield elevation), then the commanded speed is increased and flaps are retracted according to the flap retraction schedule as the aircraft accelerates.
In case of engine failure on T/O, if below V2 you target speed becomes V2, if between V2 and V2+15 you maintain whatever speed you have, if above V2+15 you trade speed for height and your target speed is V2+15.