A flow diagram would help put your idea's into something thats easier to make a circuit from. Driving lights/motors/power things is easy from logic signals, its the control part I assume you want to get right first. I'm thinking, for simplicity, a 555 timer with a 4017 decade counter. The decade counter can have multiple outputs tied together (with diodes) so that, say it changes state every 1.6 seconds.
Tying the first 3 together means this collective output would be 'high' for 3 x 1.6 = 4.8 seconds. Then the outputs that follow would be on for 1.6 seconds each. So first 3 tied together for your 'pre-stage lights', next three-5 are the cowntown lights (one at a time, moving down), the last of which can be the 'green go' light. The next output would connect to the 555 to stop it counting - so it just stops at green. Then the operator would need to hit a button to reset the entire thing.
As for a lap counter, thee are many pure logic (CMOS/HCTLL) counters on the web which may require a fair bit of wiring when compared to a microcontroller solution, but thats all they require - cheap, and cheerful. The above circuit could trogger the counter, and your IR/phototransistor pair could stop it. Dealing with two cars could be difficult, unless each is on a different track and can have its own opto sensor (just two ideantial timers, both started by the same signal, but each stopped by different beam breakers)
I doubt anyone is going to take everything you've said and take the time to create a schematic/parts list, so I fear you will have to pretty much design this yourself. However! Many here will provide guidance, and debugging problems, and answer questions.
As you've got a pretty clear idea of what you want, it makes life easier. So break it down into 'blocks', or modules. Creating the control signals is the most important thing as driving lights, be they 10mA, or 10A is relatively trivial, it just depends on using transistors/relays/FET's/triacs. As for cost, altohugh projects tend to always come up more than expected, a few logic chips, resistors, transistors, LED's can be had from even the smallest electornic store, and with Ebay's electronic shops, sahould be easy to get a big kit of all you need for next to nothing.
I mentioned a flow diagram at the begining because thats the sort of thing that not only gets engineers wet, but also is the language of logic state machines. And diagrams are king anyway
Good luck.