Hi Jon,

1. You do need to change terminals for Indigo, however you can check in your bags at the international terminal right after customs on the left. Make sure you read the signs for the transfer, I've not taken the domestic transfer shuttle for a bit and someone mentioned it had changed to depart from a slightly different location at international, but be sure where it departs BEFORE exiting the international terminal as you can't go back in.
2. I've only had paper visas but immigration is always pretty quick both in BOM and in other Indian airports. Customs can be a bit of a hold-up as they scan your bags again
3. The shuttle is free, loading luggage onto it is free, too. If anyone tells you to pay politely decline. Don't exit the terminal unless you're absolutely sure the shuttle leaves from outside. Keep printouts of your tickets as you need them to enter the terminals/shuttle (or have your e-ticket open on your phone though some checkpoints don't accept this, but I've not seen them reject it). Know the address where you're staying in India before you land as you need to fill it in on the landing card. IF you do exit the terminal and not take the airport shuttle don't take a cab, rather take an Uber. International to Domestic will run you <Rs150 on Uber.

EDIT: Taking an Uber is also not much slower than the shuttle. Not faster either though, so try get the shuttle as a first option.