Hello,
It seems that R3 actually gives the chip a Power-on-Reset. So, what is the correct assumption?
Indeed it can be confusing, but if you look at the safety manual chapter 2.7.8 The R3 fault-reaction type and 2.7.9 Diagram: FCCU fault-reaction type processes you will see:

So Chip itself cannot issue a power on reset. It can only signal to outside world that there is need for such and an SBC can then cut power for the chip or do power on reset.
However the chip itself will also issue functional reset, as this is what it can control. It is for the case where the SBC is not desired to do power on reset or will fail to react.
so that the application can attempt to recover the fault" in the R2 type?
Some faults are not critical for the system as whole and will only affect small area of usage that system can run further.
Imagine it like safety state in car, when can is still able to run, but with limited functionality so you can go to service and don't need to tow it.
Some fault can be recovered, like voltage drop on ADC for example, where it can be caused by weak power supply, or induction,etc... once this issue fade the ADC works fine again, and you don't need to reset whole system.
Knowing the Source Module i configured them to be Hardware-Recoverable and the reaction type to be a Short Reset Reaction. Is it correct or do i need to set any other configuration (Activate EOUT Pins, Enable Fault State NMI, etc)?
This is dependent on your application. Each application will have different requirements.
In general for NCF[0] - [3] you will need at least reset. As ECC fauls in flash will require re-programming of it in order to clear them.
For NCF[4] fault group the recommended recovery mechanism is "Interrupt followed by SBC initiated POR recovery initiated in interrupt service routine", so i assume it is either R1 or R2 (depending on the fault configuration) + R3, but my question is: how do i configure that in this table on Design Studio?
I am not familiar with this in S32DS, but
You simply check all options I assume. as you need EOUT to be active. Also Alarm with timeout, and the NMI.
So if you are able to solve it in ALARM interrupt, then you will recover from there. If not then EOUT will signal out the issue and SBC will take actions.

Best regards,
Peter