Timing when clocking MCU off of A/D
The inverter between the ADC and the F/F is provided in the original application note, but I don't understand what it's there for. The ADC outputs have over 4ns of set-up time before DRDY goes high (per the ADC datasheet). The maximum setup time required by the latch is about 3.2ns. So, probably the inverter is not necessary, but it doesn't hurt anything. All that the inverter does is to vastly prolong the setup time for the latch. The inverter may act as a clock buffer, but a non-inverting buffer like 74AC125 would do the same job and still relax the set-up time constraint: it would delay DRDY, after all. Both a non-inverting buffer and the inverting 74AC04 (not ACQ - that's a typo in the datasheet) would add between 1ns and 10ns of propagation delay, so all's good - the latch's setup time specification on the data input is not violated.
The MCU can be fed the same clock as the latch, but it will need to trigger on the falling edge (opposite from the edge the latch triggers on), so that it will have enough hold time. Triggering on the rising edge would probably violate the hold time. This paragraph applies whether the buffer between the ADC and the latch CLK is inverting or not. I also presume that the MCU is using DMA to capture the data - you should check that the DMA bandwidth won't deprive the rest of your application from memory bandwidth.