Oh well... I got the answer the hard way. After spending an hour replacing the busted alternator with a new one from Duralast, I figured that (1) the old malfunctioning alternator had destroyed my brand new battery, which had to be raplaced, and (2) they sold me a deffective alternator!!! This one outputs 12.5 volts at the terminal, when A/C and headlights are on - waaay lower than 13.7V should be (on my 318i).
1) There is a free alternator testing machine at Discount Auto Parts, which tells you whether your diodes are gone, or something else.
2) ALWAYS go with the BOSCH-rebuilt alternators. It's cheaper in the long run.
P.S.: While removing the stock air box, I discovered that it has a lining - sort of like another air filter - on the bottom, top and sides of the inside of the airbox. Where would I get a replacement for it, and what's the proper name?