Black mold under the bathroom tiles?

We rent a home that was built in the 60's, and from apparent years of coats of paint on the bathroom ceramic tiles, the paint just kept chipping and peeling off, and getting everywhere, especially in the tub when my daughters would have their baths. We tried several different methods of paint removal, but there were too many coats, and it just barely came off. Our landlord agreed that we could remove the tile, and put up a tub surround. In beginning of removing the tile, we came to realize that all of the sheetrock was damp, and all along about the bottom foot or so of it was rotten, and crumbling into mush, and some parts of it had a black color to it. A little bit further into the project, i began to feel physically ill, almost to the point of vomiting. We've lived here a year, and i have noticed that mine and my daughters allergies seem to be getting worse also. Should i be concerned that this IS black mold? Does the entire bathroom need to be torn apart?