Hi David M
I run R/C drag cars too
1 the prestaged and stage lights light on when the car is go to the line detected the tire (Body in RC) by the photocell, photoswitch etc not mechanical switch.
2 the sequential count down time for 3 yelow bulbs to green is .5 sec for normal or bracket racing 1.5 sec total no 3 second, and .4 for "Pro light"
Full tree
3 for clock you need a .001 resolution timer when minimun of 5 digit 9.9999 no minutes timer !!!
4 the timers start when the car move in the line when the staged light is off
if you only need this, you make it whit electronics circuits, but if you need more Reaction time, Speed, Handycap 6ft, 33ft 1/8 time you need a microprocesor, memory, eeprom, programing project.
I Design and build similar circuit in 1981 for Scale Drag Racing, and at present work whit it, maybe I can help you whit some circuits !!!!