The back hazard lights (which are reverse lights) should be connected to a small switch that screws itself into the gear change joint below your gear stick; it will have two wires coming out of it, and this switch basically shorts your indicator wires when it is pushed in (by putting the gear in reverse). I'm not fantastic with electrics, but there is not too much to the reverse lights, so maybe your old switch is gummied up and stuck on the on position (by grease and dirt etc). Try unscrewing it out (if you can without hurting the wires, if you are capable, removing it may require cutting the wires, but only do this if you are able to crimp them up nice and neatly - no electrical tape jobs! Especially on those wires which are subject to the elements) and seeing if the small button tip is stuck in.
As for not kicking over, if you can get your hands on a compression tester or a mate that can help you out, try starting there. Also, if you are getting spark it doesn't sound like an electric problem? Try getting a can of "Start Ya Bastard" read the instructions, and spray that down your carb.
Here is a list of possible reasons your car may turn but not kick:
- Ignition system damp or wet
- Ignition leads to spark plug loose
- Shorted or disconnected low tension leads
- Dirty, incorrectly set, or pitted contact breaker points
- Faulty condenser
- Defective ignition switch
- Ignition leads connected the wrong way round
- Faulty coil
- Contact breaker point spring earthed or broken
- No petrol
- Vapour lock in fuel line (in high altitudes or hot conditions)
- Blocked float chamber
- Fuel pump filter blocked
- Too much choke, allowing too rich mixture to spark plugs
- Float damaged or leaking or needle not seating
- Float level incorrectly adjusted
This is quite a long list, taken from the Haynes manual, but each item is quite relevant and should take 10-30 minutes each task, so slowly spend a weekend going through and checking each item off. Let me know how it goes.