Quick charge adapter on not supported devices
Is it safe to use it with my old phones? 



It is safe. See what happens if you plug a non QC 2.0 device into a QC 2.0 charger?



  And what would happen if I use that adapter with devices supporting fast charging technologies from other companies.



They all are proprietary technologies which vary in terms of what is changed to charge fast - Voltage, Current or both, apart from other factors. Details obviously are not in public domain. Therefore, they are not guaranteed to work on devices using other technologies, so one cannot predict whether it works or not. 

OEM's don't recommend such practices. YMMV