SOLVED: Just an update of what I've done.
RECAP: Computer has restarted randomly over the last couple of weeks, and now computer will not boot at all. When powered on, fans will spin, storage devices will make sounds, but no video output will come on the screen.
- removed the GPU and used the motherboard video output,
- tried two separate video outputs (HDMI and DVI, both of which work for other computers),
- moved the 2 memory sticks into every combination of slots on the motherboard (Z-170A PC MATE),
- cleared the CMOS,
- moved 2 freshly bought memory sticks into every combination of slots on the motherboard,
- bought a new motherboard (Z-270A Pro),
- bought a new PSU, AND
- consulted a University tech center that stated it's probably the processor at fault. I bought that processor in October of 2015.
I ordered an Intel i7-6700k. Swapped it in. Computer's working fine.
Hope this is helpful for future issues. And thank you to those who helped.