My question is: 1. Can my father continue the same way of issuing the final installment from cheque and give affidavit?
Your father can continue to make the payment in the same manner as was done earlier.
2. Or Should I get my father a PoA for this plot by going to India first and then let my father pay it using his cheque.
What is POA doing in this. If you want to to authorise your father to sign the papers for purchasing the property on your behalf then you can authorise him by executing a POA deed in his favor.
3. Or should I open NRE account and can let my father pay part of final amount from his cheque and rest part of final payment from my NRE account?
It is always advisable to open a NRE account in India not only for this purpose but it will be useful for some other purposes in future too.
Also, will I have to do anything additional during registration of this plot like certify with RBI or something. I am sorry I am not aware of this formality.
The permission or any certificate from RBI is not necessary for this situation.