Bad connection at battery, bad connection at starter, relay problem, defective starter, wiring issues. More symptoms would help eg: does it turn over but won't start? does it click the starter solenoid? When you turn the key do you get power to the relay? Also check the fuse box in the right rear corner behind the battery. They are known for the body work rotting out and the box coming loose causing electrical problems
Check the charging system with a voltmeter to see if it is charging the battery sufficiently A fully charged battery should read approx. 12.8 volts without engine running With engine running should read approx. 13.8-14.2 volts
If the battery has a defective cell it might not allow the car to start with a charger. Have the battery tested.
No you do not need a battery, but the lights and horn will not work unless you have a battery installed, dead or not.
Have the battery load tested at your local auto parts store, even though you think you have it charged it may be bad. Have the charging system tested also.
you should be able to start your jimmy up right after you install your new battery...if it doesnt, I would replace or clean your battery cable terminals and see if that works!
Check the fuel pressure on the Fiero to start with. It sounds like it could be starving for fuel.
Do the headlamps work while cranking the starter? If not focus on your battery and cables. Jump start the car and have the system tested before you throw more dollar bills at this problem; it could be the alternator but have it tested first.
Sounds like the battery is dead or you have a loose/dirty battery connection. Have the charging system and battery tested at your local auto parts store.
Use battery load tester and properly test battery. When battery fails, replace it.
I assume you had the battery properly tested under load. Its starts and runs when jump started so that tells me assuming the battery is in fact good that the battery cable connections to the battery are bad and need cleaning.
it might be the alternator, try and get it tested. if the car won't start the battery will also be dead, you could replace the battery and drive it to get the alternator tested at an autozone or advance auto parts for free.
check your main fuses