Customs at Indira Gandhi Airport, New Delhi is fairly fast, especially in very early morning in your case. I have been done International to Domestic Transfers in 1-1.5 hours. Since yours is on non-peak time, it should be even faster for you.
Unless your incoming flight is significantly delayed, you don't need to worry about this.
In India you usually need to show your ID at two places - entrance to the airport and at check-in.
As per this news item, Bureau of Civil Aviation expects passengers to have any one of the following IDs to enter the airport:
Aadhaar or m-Aadhaar
Student ID card
Passbook of account in ...
As far my knowledge, the arrival and departure gates for domestic flights in Mumbai Airport are nearby. However you will have to go through security checks again at departure gate. So, to answer your questions:
If both the flights are operated by same airlines (in your case GoAir) and you have a single ticket for your journey, you don't have to worry about ...