A lot depends on the reason for the leakand the type of policy your mom has. If the leak occurred as the result of no maintenance or benign neglect the most likely the cause of the leak nor the damaged caused by the leak will be covered. If the leak occurred as the result of some sudden and accidental damage then most likely the leak will be covered.
The asbestos issue is complex. If the repair of the water damage does not entail tearing out any part of the ceiling then the asbestos issue is of no consequence. If the repair entails removing any part of the PACM(presumed asbestos containing material) then the policy should cover the increased cost of dealing with the asbestos. This is similar to sustaining damage to a floor that contains asbestos and now has to be removed. Most policies will pay the additional cost to remove and dispose of the asbestos tiles as well as the air quality monitoring while the abatement is underway.

Ask your insurance agent, that is part of their job.

She could check with the contractor that did the installation. Typically they cover workmanship for shorter period of time. If it was due to a defect in the shingles then the shingle manufacturer may cover it.

Your best bet is to check with the insurance agent, as this would be the easiest and require least amount of work on her part.

I seriously doubt that the leaky roof would be covered by her homeowners insurance, unless it was somehow leaking due to damage from wind or hail or possibly a tree falling on it, something like that. The consequential damage of the ceiling from the leaking roof will more than likely not be covered, for the same reason.

Your mother should check the paper work and guarantee that she would have received when the roof was put on. Most reputable companies will warranty their roofs and the labor for normally 20 years. There may be a provision in the agreement for collateral damage caused by any defective shingles or installation.

Shingles are usually rated in weight, and years of service. A 20 year roof is only as good as the installer. If the shingles curl, or break in the 20 year period, you may be able to receive money back from the manufacturer. I would contact the people who installed the roof and see of they will make the necessary repairs.

Your best bet is to call the insurance company

tell her to call her insurance agent.