I made two prints from the center of the stack of paper. There are still spots, but they aren't as bad as the first one, which has been the worst by far. Nose oil completely got rid of the scratches, though which makes me happy.

That looks like a flaw in the negative that doesn't show up with smaller enlargements. I wouldn't worry about the paper so much. One things for sure, it's not a light leak.
It can't be on the negative, because I've made 4 prints now and the spots are in random places. My darkroom door isn't perfectly light tight...I can see a bit of light in places around the edge. But I consider it light-tight enough to work with paper in there in the daytime. If there were light leaks, I don't see how they could be focused onto an image that would give small spots. I would expect general paper fogging. And besides, the worst print so far was made last night when it was dark even outside the darkroom. I made the exposures through an Ilford under-lens filter, but there doesn't seem to be any pinholes in the filter. I suppose a tiny pinhole could expand into a 3mm light spot on the print by the time it got down to the baseboard.