battery light staying on with the engine running normally points to a problem with the alternator. get a multimeter (voltmeter) and measure the voltage with the engine on and the engine off by put the prongs to battery terminals. voltage with engine off should 12ish, voltage with engine on should be 13.5-14.5 or there abouts.

its probably due to the voltage being too low with the engine running, because of a faulty alternator. the grinding noise could be something to do with it. maybe theres an issue inside?

might be best off slapping another alternator on and seeing if that changes anything.