My engine light came on

Check your fuses.

Alternator. Have it tested with a voltmeter while on the car and running. Or take it off and have it tested. The current to run the car and the accessories is from the alternator when the engine is running. The battery light coming on alerts you that the alternator is not doing its job.

i see dash lights on when you try start engine.could have faulty starter relay or bad starter solenoid or starter bad.use a 12 volts test light see if power going to starter.if starter circuit okay could have theft security problems because BMWS did have theft security key ignition system back in 1994.

Light bulb ? Fuse?

what do you mean a new key?
changing keys on this car will require you to have the key programmed to the ecu. only bmw can do this... they charge $100, they need the car with the ecu and the keys...

Possibly. 1st have the battery checked though. Pull it out and have a parts store check it (free service). All the lights may work, but the battery may not have enough 'juice' to crank the engine over. Is there a click, or no sound at all? Check terminals and connection at starter also to make sure all are clean and tight.
If battery checks out as fully charged and ok, then pull the starter and have it checked out. Good luck.

Sounds like the alternator may be missing an excitation current source.
Check the 'IGN' dash indicator lights when you turn ignition switch on prior to starting motor,
if not one of the below may be problem:-
1) check exiter control wire is connected to alternator D+ terminal (low current wire, +V with Ign on),
2) replace dash 'IGN' indicator lamp.

It sounds more like the alternator is bad. This will cause the battery to run down.

Do you have to pull something to turn on your headlight?
If yes, then don't pull it all the way back. You can pull that thing three time.
first pull will make you speedometer and all other things glow
second pull will turn on the HEADLIGHT
and the last (i.e. the third pull) will make your High Beam turn on
Hope this help, I have a 1996 BMW too.

hello, check for a damage crank position sensor this sensor sends the singnal to the dme (digital motor electronics ) to inject fuel and pulse the ing coils and will also start the fuel pump when it see a rpm above 500 this is a good tell sing if you crank the engine and the rpm does not jump there is a good chance that the crank posistion sensor is bad / the battery and engine light will be on untill the engine strarts