A high quality hepa filter will remove airborne spores. That said, I don't worry about mold spores during demo unless the concentration is really heavy (then I wear a mask), but I don't have any allergies so YMMV.
I actually prefer to tear-out moldy material when it's damp - it's easier to demo and makes for less dust. Then I'd put one or two fans on what's left of the ceiling framing 24/7 until it's bone dry.
The next step would depend on why it got moldy in the first place - if there's an unresolved water issue there's no point putting up a new ceiling since it will just get moldy again.
Texas went thru an era of mold remediation that cost the insurance companies tons of money until the State got involved. There is no such thing as a Lazarus mold. Once it's dead- it's dead. Unless the drywall is beyond repair you can treat the area with something like Microban which kills the mold and any spores. Then encapsulate it with something like Kilz oil based sealer. You don't have to remove the drywall if it's in good shape. But the key is finding the source of the problem and addressing it first.
No don't put a fan on something that is already moldy. Tear it out. Drywall is cheap and expendible. Any drywall that was wet enough to grow mold, was wet enough to be structurally deficient anyway. Solve the moisture problem first. Take the wet drywall down. Dry out the framing and inspect it. Treat it with a fungicide. Dry it. Seal it. Replace the drywall. A good HEPA vac will pick up mold spores and trap them. A good HEPA vac is usually not available to the average DIY homeowner. Regular junky vacs with HEPA filters are. They are not the same thing. All is for nothing if you don't solve the moisture issue. Never heard of Lazarus mold either, but mold that isn't exterminated by a fungicide will go dormant when moisture levels drop, only to reactivate if it gets enough moisture at some point in the future. Sometimes years into the future. Dead is dead, dormant is not dead. Fungicide makes it dead. Drying it makes it dormant.
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