1. Do i still need to carry a printout of ticket for security? - No, it has been ages that I have actually carried a print-out of tickets to be shown to CISF (security agency)/Check-in counters - Copy on Mobile Phone OR Airline/TA mail works perfectly fine (IN your case however you would have both of your BPs issued from EWR (EWR-BOM & BOM-MAA) so just show BP to enter T1B)

2. Do I need to clear customs and immigration in Mumbai? If so, seems like I should carry my check-in bags from international terminal to domestic? - Yes, although your luggage would be tagged all the way to MAA but you would have to physically collect it at BOM T2 (INT) and hand it over to the 9W counter just right across and proceed for catching the BUS for terminal transfer as your domestic MAA connection would be from T1B.

- I would recommend to keep a pace as it would be peak time with multiple long-haul flights for immigration etc and also loads for terminal transfers - Hand your luggage fast in T2 and get into T1B ASAP. Seek the help of 9W Kids around T2-INT, if any.