After I zoomed in on pic 5, It's possible that some water could be getting through the oakum. Not likely. It doesn't look like sewer to me. The black sticky substance could possibly be some type of reaction with the pipe coating/lead joint from the leaking water coming down the roof vent as you mention. Remember, those old black cast iron pipes are coated with a heavy coat of asphalt type paint.
I agree with you, it could be coming from above as in a leaking/cracked roof vent. They do eventually need replacement or repair.
Your idea of testing is sound, wait for the next heavy rain and keep an eye on that spot. Put some paper towels or bucket there. If you notice it more with a heavy rain then your theory is correct. Just my opinion.
If you do decide to remove and replace it, as mentioned, be very careful, it's very heavy. Do it in sections with braces.
Hope you get it figured out and everything working the way you want Tom.