Does anybody know if Riposte and Retribution proc during channeling Warpath?
Yes, they can, though Riposte has a short duration.
So it has to be manually casted when it procs or does it automatically cast the attack on the backend?
Riposte is a proc/buff from Vengeance that has a duration of 2 seconds, so if you use Vengeance & then immediately start Warpath-ing then the first time you get hit within 2 seconds of using Vengeance Riposte will proc.
Retribution is from a passive that can be proc’d any time you get hit.