The engine management light is usually a yellow one, the charging one is a red one.
Flat battery shouldn't cause higher idle speed. If it's totally flat than the ECU loses it's learned data and defaults back to factory settings, which can make the it a bit less smooth to drive while it get it's bearings again.
If the alternator is broken, the battery will stop getting a charge and the car will either refuse to start or misfire for a while then grind to a halt while driving it. No electricity = no spark.
Higher idle speed is usually something telling the engine it's colder that it really is so it does the electronic equivalent of pulling the choke out, or a duff Lambda sensor (exhaust emissions) which has the same effect.