To add to PappyJ's post, in summary -
Are climb gradient requirements for aircraft certification or requirements that an operator should comply? or both?
The Design Standards require minimum still air, out of ground effect, climb capabilities depending on number of engines. These minimum limitations (generally referred to as WAT [Weight for Altitude and Temperature] limits) give you a reasonable probability that, in the event of a critical failure, the aircraft will continue to climb. Note that turbulence and inversions are outside of the square and may cause you to go down rather than up.
On top of the WAT limits, the operator/pilot may need to restrict TOW further to achieve other requirements, such as runway length or obstacle clearance gradients.
Where these numbers (2.4%, 2.7%, 3.0%, …) came from?
The history will be in some ancient ICAO document which I am not able to cite for you. In the nature of ICAO matters, Signatory States then implement ICAO requirements in National requirements. So, for the USA, FAR 25, for example, imposes the WAT limits on the designer for certification purposes.
why the climb gradient for the second segment is 2.4% and not 1.2% (twin)
The original Standard will be based on statistical work resulting in probabilities which meet the normal design philosophies, ie a very small probability that a failure will result in an accident. A twin will have imposed a shallower gradient than a three or four motor machine as the comparative loss of performance is much higher .. for example, considering that climb relates to T-D, loss of half the thrust may result in loss of, say, 70-80 percent of climb performance for a twin.
why the second segment is usually the most limiting than the third or the fourth?
Not necessarily the case. However, given that, once we are up and away, the main concern is terrain .. and that the terrain clearance calculations are based on net performance (gross, or reasonably expected performance less a fudge factor ... 0.8% for twins), terrain generally becomes less of a routine worry the further we are away from the runway. Also, for the majority of runways, most of the terrain problems are reasonably close in to the airport.
If a turn is required after engine failure during takeoff, does this affect aircraft performance
Certainly does. Depending on the aircraft polars the typical climb gradient decrease for a 15 degree bank turn will be something in the region of 0.5 to 0.9 percent. The general restriction to a maximum bank of 15 degrees is due to the significant ramp up in climb performance loss as the bank angle increases. Note that, for some runways, the procedure may require a smaller angle of bank to achieve a larger turn radius.
small/big radius than usual?
variation in turn radius will depend on speed so the range of V2s has to be considered in determining the turning trapezoid area to be considered for obstacle clearance.
turn into the inoperative engine or into the operative engine
not relevant unless you are going somewhat slower than you should be. Main concern with which side the failed engine is on relates to Vmca considerations where the real Vmca depends very critically on bank angle.
when those certification criteria can not be met, the airline is required to publish a proven alternative stratagy
Not the case. The WAT limits MUST ALWAYS be met in the RTOW for the runway on the day. However, if the straight ahead RTOW is not adequate for the airline's commercial desires, it is perfectly appropriate for the airline to investigate turns to see if they can achieve a better weight. A case of matching the loss in climb performance against a possibly better terrain profile and it doesn't always work .. ie sometimes the best weight is straight ahead and you just have to accept whatever the RTOW is. I can recall spending a GREAT deal of time trying to get a better weight out of Gladstone (Queensland) years ago and all I achieved was a lot of practice at drawing curved splays .. at the end of the day I had to admit defeat for the aircraft concerned and go back to the original straight ahead calculations.