Please post the HP part numbers for the ICs and transistors. Maybe I (or someone else) can cross-reference them to standard parts.
This looks like a very standard digit scanned display. All segments of one digit are driven at any time, and the system simply repeatedly scans through all digits fast enough for your persistence of vision to create a steady state display of all 8 digits.
The digit to be displayed at a given instant is determined by U56, operating as a 3-to-8 line decoder, for which a single output is driven LOW. A quick google reveals that the 74xx155 is a very likely candidate for this, as the pin-out and functionality match well. It is a dual 2-to-4 decoder that can operate as a 3-to-8 decoder.
The LOW output for the digit to be displayed will turn on the darlington transistor, providing +5 V power to the common anode of the selected digit (for digit DS1, +12 V). The individual segments are driven by the combination of U62 and U71. Realise that even though these drive the segments for all digits, only one digit is powered at any given time (via U56 and the darlingtons). U62 is a 4-bit BCD to 7 segment decoder, probably with open collector outputs, since the NAND gate U46c is wired directly to one of the outputs (U46c drives the center horizontal segment of the display "-" when a code of 0xC to 0xF is present). For each 4-bit code between 0 and 9, U62 drives the appropriate segments to create the numerals "0" to "9". Since the +/- digit DS1 requires 12 V power, then U71 must have open-collector outputs. U71 is likely a darlington transistor array, or a bus driver with OC outputs.
You are right to be suspicious of Q16. Check whether its output even goes to +5 V like the others do. Also check that pin 10 of U56 ever goes low, and check R65.