If the vehicle is charging OK (as shown by the alternator test) then just get a new battery in there before you get stranded again. Do you know when it was last changed, or if it is original?

The grinding sound doesn't appear to be related to a belt, from your description. More likely to be in your power steering. Get it checked out.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue