What did an arriving northbound Harlem train do? Did it continue over to the B&A and make a reverse move into the station?
I can answer this for the 60s, and I do not think this practice began in the 60s. The answer to the second of your two questions, quoted above, is "no." Harlem passenger trains did not go onto the B&A mains.
The answer to your first question is essentially contained in your third question which is:
Was there a platform opposite the station on the Harlem line?
Yes, Harlem trains pulled up to the east end of the wye track where there was a short --- one or two cars long --- wood platform on the east side of the track. So, passengers debarked on the right side of the train, and walked around the front of the locomotive and across the B&A tracks to the B&A station.
I remember power running around the trains, I did not see a whole train turned on the wye any time I was there personally. although I will not assert it was always done this way. When I rode up there it was on one of the weekend trains that had two diesel units (RS-3s) and they pretty much always coupled the units short-hood-to-short hood, so when they ran around the train the engines were oriented correctly, i.e., NYC RS3's cabs were oriented to run long hood leading.
The weekday train*** had only a single RS-3, and I sort of think they would wye the power to keep it running in the preferred direction.
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*** And the conductors who owned that weekday Chatham train were some of the most fascinating and delightful railroaders I have known. Since I'm talking 40 years ago, and these guys were wa-a-a-y up the seniority roster then, I'm sure they're long gone now: John "Foxy" Fox, John Mallon, and Albert "Salty" Wilcox. God bless 'em all!!!