What is in century old chimney dust, and what else can I be doing?

Sorry I can't give you any info on what health concerns you may have from the chimney dust, but I can tell you I have removed a similar chimney in an older home. Even though all the carpet was removed in that room and all doorways had plastic covering them the dust was unbelieveable. Cleanup took a lot longer than removing the chimney. Be sure to keep a detailed account of the time spent cleaning, tell the manager how much time you have into cleaning already and ask what the compensation is going to be for your time and when will their cleaning crew be showing up. I would also be concerned about anthing electronic.You may wish to have TVs, computers, electronic games, etc. professionally cleaned. If you have renters insurance, contact them. They may go after the company that caused the damage for you.