I am using my Pi's GPIO to control some 12V lighting strips (LED). Can I just ground the GPIO connection to the 12V supply's ground? Or do I have to keep the RPi's connection grounded back to the board? I have heard that using 12V to GND 3.3V can create negative voltage, but I don't know.
The Pi's Gnd (0V) should be connected to the Gnd (0V) that 12V is "referenced" to (just like the Pi's 5V and 3V supply rails share a common Gnd (0V)). However, make sure that none of the GPIO pins can connect to the 12V ie. they should be isolated via a transistor or other buffer circuit.
Still running Raspbian Jessie or Stretch on some older Pi's (an A, B1, B2, B+, P2B, 3xP0, P0W, 2xP3A+, P3B+, P3B, B+, A+ and a B2) but Buster on the P4B's. See: https://www.cpmspectrepi.uk/raspberry_pi/raspiidx.htm