Old tin pans will turn black and will turn other things black if haven't been washed for a while. They were not made in China. I have a food mill that I need to wash just before use or it will turn things black. If I used it yesterday, I still need to wash before use. Whatever that black is, it will not hurt you. It's just unsightly. I would cut off the black part, eat the cake, and wash the pan well before each use.
The exploding dishes were the pyrex type that were made in China. So don't put those dishes in the oven unless you know they were made in this country.
And remove your baked goods soon after baking. Do not store leftover cake in them.