The 2n3904 transistor converts the varying output voltage of the op amp to a two state logic level signal suitable for input to the NAND gates. The switching point between logic levels is determined by the combined settings of the DIR CAL, DEAD BAND, and GAIN pots and the circuit input level.
The pair of NAND gates immediately to the left of the 194 shift register control the mode of the shift register (either shift left, shift right, load, or do nothing). The single NAND gate to the left of that pair ensures that one of the inputs to each NAND in the pair is always of opposite logic state so that the shift register will shift right or shift left depending on the logic state at the output of the transistor. The outside inputs to the pair of NAND gates are connected to the appropriate outputs of the shift register to generate the desired shifted pattern of LEDS.
The precision full-wave recifier is a valid textbook design, except I don't know why two diodes are used in series in each leg, instead of the usual single diode in each leg. Perhaps you are not simulating the entire circuit which is composed of the two op amps on either side of the diodes, the diodes, all of the 220k resistors, the voltage from the dead-band control, the voltage from the zero control, and the voltage from the left-most op amp.