1 - why should I cook sugar and water until 121C and add it into egg yolks while it`s beating slowly ? What`s the chemical process inside this action and it`s purpose?
Well basically, heating sugar to 118C to be exact (soft ball). When sugar reach certain temperature it change its properties, hence getting chewy/ firm/glossy meringue at the end result. There is alot of terminology for each state (temperature reach) soft ball, firm ball, hard ball, hard crack. Different stages has different result and usage. And why not purely egg white? because adding hot sugar into egg it kill off bacteria by temeprature.
Refer over here for your reference http://www.joyofbaking.com/StagesOfCookedSugar.html
2 - why the cream should be cold if my intention is to do chantilly?
Chantily and Meringue is total different thing, Chantily is whipped cream with icing sugar. It is store in chilled temperature because cream is fats, and fats is volatile/sensitive to temperatue. so it must be kept chilled/pipe chilled
3 - why we need to emulsify sugar and egg whites in the water bath till 50C when doing french merengue?
French meringue? water bath? i think you are referring to swiss meringue, it is to sterilize/ kill off bacterial by hitting certain degree C, and cooking egg white/egg yolk it is much more stable compare to french meringue which use raw egg