I've flown EWR-BOM in Y 5 times in the past year and a half - and yes, as others have said, the flight has been full every time with people gate checking bags like crazy. I like the flatbed CO seat myself, and the fact that you avoid connections and get to where you need to go (or in your case, almost there).
Upon arrival in BOM, it's not so horrible, just bus it from the plane to immigration and then customs. When leaving customs, it can be a little daunting with all the people waiting outside with signs - just be patient and look closely for your pre-arranged driver/car.
A hotel in the area I stayed at before that was good was the Intercontinental Lalit hotel. I believe now, unfortunately, they dropped their relationship with Intercon and are now part of the Lalit family of hotels in India. Either way, it is the closest airport to the international terminal and offers car service to pick you up.
Never flew domestic out of BOM, so can't help ya, but you can just then take the flight the next morning and have the hotel drop you at the domestic terminal.
On the return, you will have access to the LH lounge with a C ticket or *G and Y ticket. The LH lounge is small, but has some decent finger food and drinks. As is typical with flights to the US, there is another security check at the gate, so ditch the liquids and be prepared to queue once again. Also, as others have said, make sure not to arrive more than 3 hours ahead of your flight departure, and have a printed itinerary to show the guards - if either of those is not met, they won't even let you into the airport.
All things considered, if this is your first time in India, I'd follow chinatrader's advice and see about connecting in the ME instead. Well, anywhere except Jeddah... I just home a few hours ago after a 37 hour trip SIN-RUH-JED (South Term)-JED (North Term)-FRA-EWR. The worst part of it was the connection in JED, which makes BOM look like an organized, well lubricated machine.