Well you need to make sure your getting gas to the engine. If you can, have the fuel pressure checked closer to the engine, like the fuel rail. Then make sure you have a spark at the plugs. (safely) If you've got gas and spark, the only thing you're missing would be air. Check the vacuum hoses, air filter, and the throttle body. Best of luck. Most likely it's an Ignition issue. The first thing I would look at would be the Ignition Control Module "little gray rectangle box attached to the distributor that has a small wire harness plugged into it".
Electrical engineering is passing through an interesting stage and there are so many new technological things happening, advances in alternate energy etc and it seems to be an exciting field to work and develop new ideas in.
battery low on charge, or the battery is bad, or a faulty starter, just google, how to change starter on E46. for instructions ( written by me ) and you will find them.
possibly the ignition switch electrical part , but you will want to pull the key cylinder and make sure the little piece on the back of the cylinder is not cracked off nomal indicated by a key that seems to turn to far
bad electrical connection.some where.
It seems that the carburetor is misadjusted.
faulty sensor possibly cam/crank/oxygen/temperature sensor.
It seems you have an electrical short somewhere.
Mine went at 38,000 miles, seems to be a commonly replaced part.
1. The economic idea that seems to be the most widespread is the idea of capitalism
could be an overcharging alternator//seems to happen more with 'cheapo' batteries like maximo... need to get a proper AVR test on the electrical system from a competent shop.
bad fuse, relay, wiring, or motor... get a test light and start trieng stuff
do you mean a lightening bolt symbol? Seems to be an electrical indicator.