The left transponder is usually the primary transponder.
Not necessarily. The "primary" transponder is whichever transponder is selected (ie L or R) during the preflight. That selection is airline dependent. At Malaysian, the SOP (as stated in their OPS-A) was to select the L transponder on outbound flights from KUL and the R transponder on inbound flights to KUL. In this case, the L transponder should have been selected given that MH370 departed from KUL.
The air data for the left transponder is from the left AIMS cabinet.
The transponders receive air data from BOTH AIMS cabinets. The L transponder receives ADIRU data from the L AIMS cabinet on ADC Input1, and SAARU data from the R AIMS cabinet on ADC Input 2. The R transponder receives SAARU data from the R AIMS cabinet on ADC Input 1, and ADIRU data from the L AIMS cabinet on ADC Input 2.