I am having trouble putting all your symptoms together. For the alternator light to be on with the key off you almost have to have a bad alternator, BUT! the alternator has nothing to do with starting the car, SO, maybe you have an ignition switch problem.

Its possible that you have a ground problem either with the alternator or engine also. Your running problem with accessories and a good battery almost confirms a high resistance circuit in either power or ground. Get a voltmeter and take voltage drops acroos the circuits you are working with.

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Steve Brotherton
Continental Imports
Gainesville FL
Bosch Master, ASE Master, L1
33 years MB technician