HIGH IDOLING GOOD & ALTERNATOR STILL BATTERY DIES

Have your alternator check out!

A battery place can test the battery and the alternator to see which one is faulty

otherwise get a volt meter and check the voltages from the alternator without the battery being connected it will be higher at approximately 13.8 volts and from the battery 12 Volts .

I would not drive it in case the engine died, the symptoms you describe are indicative of possible cylinder head issues.
Get a GOOD garage to visit or have them tow it into their premises to check your vehicle over.

Check fuel, spark plugs and wires.

Possibly a loose or corroded connection somewhere, or bad alternator, not charging battery and caused system failure. Check for good clean connections at battery and starter. With a volt meter a battery should read 12.4 - 12.6 at rest, if you have a functioning alternator battery should read up to 13.5 to 14.5V .

you have damaged the alternator. running the engine and light etc on a bad battery will burn out the alternator. you will need a new one or one that works also check the wires to the alternator as you may of burnt the wiring aswel.

I would test alternator again but I have always had trouble with the machine test so a good way to test is to start the car and disconnect the positive cable from the battery, if the alternator is good the car will stay running, if it is bad the car will die.

You need t have your charging system checked. Low voltage can affect the engines sensors that monitor engine performance. If the battery isn't being charged while you are driving, it will eventually be drained and the engine will die and yu will lose power steering and brakes. If this happens in traffic or at high speed, you could be in big trouble. Have your alternator checked or replaced. Wouldn't hurt to have the battery checked at the same time.

check to see if your voltage regulator is in the alternator or on the side wall. a bad voltage regulator will keep draining the battery.

Your alternator should put plenty of voltage out at an idle, you might take it out and have it checked.
Most local parts houses have the equipment to check them without charging you.
If the alt is not working your car will run until the battery goes dead and alot of cars wont run without a charged battery and alternator not working.
You might also have a problem with the voltage regulator, or I should say that that is where your problem may lay.
Good luck