Varadh

I will be traveling in business class from Washington to chennai transiting at Mumbai by Lufthansa and Air India in one PNR.Arrival at Mumbai is by LH 756 from Frankfurt at 01.00 hrs and leaves @ 6.15 hrs Mumbai by AI570. I understand that The Air India flight from Mumbai to Chennai starts from international terminal. I have few doubts. Can my boxes booked directly to chennai at IAD? Where do I undergo immigration formalities? Chennai or Mumbai? If I have to do customs check at Mumbai should I need to take out my boxes and recheck in? At Mumbai airport whose lounge (Lufthansa or Air India) I can use? As my age is catching up, i am little anxious. So pl don't mind to help me