During bridge removals, we sometimes have to remove asbestos based paint. It's not overly hard, it's just a tedious process. Right personal protective equipment. Sanders with HEPA vacuums. Proper containment. Paper work trail showing proper handling and disposal at an approved waste site. Landfill & hazardous waste disposal fees.
Of course, a bridge is a far cry from a home. Closest I got to that was a multi-story condo site. Installed drywall had gotten wet and had started developing mold. Complete removal of the sheetrock, PPE and disposal was required. Mold was minor and we was able to Kilz over the studs. Again, not hard work but tedious and scrutinized by all the involved parties.
Assuming your asbestos test comes back negative, I'd get some PPE and get the mold removed & disposed of as quickly as possible. Once you get the studs exposed you can make a determination how bad and far reaching it actually got and what steps you will have to take to remedy the situation. You may require some reframing depending on severity. I personally don't like the Kilz approach and would replace the affected framing in my own home if it was reasonably doable.
I am not familiar with the removal & disposal requirements for your location. You should investigate to ensure you do it right so it doesn't fall on you later. By chance, does your insurance policy have a rider for any such things? I seem to recall mine having something along this lines but I'd need to read the fine print.
In regards to your bigger problem -- a 1905 home is awesome in many ways. However, its 117 years old and has seen some wear & tear. Plus standards and codes have changed. It wouldn't surprise me that each time you do more than a simple cosmetic repair that you open a can of worms of some magnitude. Over that span of time you get a variety of homeowners with various budgets, skill levels and different mentalities about repairs. With home prices up, one solution might be to make these repairs and then get the house ready to sell and then repurchase a newer/different home.