Hi qw,
The ADS1298 device should not get 'a little hot' - is you reference slowly declining as the temperature of the device increases? I suspect you've damaged the device somehow. This is not normal behavior. The /CS signal needs to be low through an entire command cycle. If you do SDATAC, dropping /CS and sending the command is fine. The RDATAC and data retrieval can all be done under the same /CS cycle - you do not need to raise it in between.
The internal DC tests depend on which channel you are looking at, this is described in the data sheet at the top of page 24, channels 1, 2 and 5-8 are 1/2 AVdd; channels 3 and 4 are 1/4DVdd. The major influence here is the reference and the quality of your supply voltage.
Both interrupt routine are located in t1298_ob.c. For the clock and data - you mentioned very early on in this thread, you had the ADS1298 working with STM32 via SPI - there really is no difference here with the McBSP. In clock stop mode, to generate clocks you must put something in the transmit buffer of the master controller. To simply generate 24 clocks to read data, load the transmitter with 0x000000.