With a tree falling on a roof there isn't much interpretation and should be an easy claim to file that wouldn't be denied.

Any claims for monetary reimbursement for medical expenses due to illnesses, discomfort, or medical conditions that the tenants may claim were either caused by or worsened by alleged mold in the home will be denied unless the claimant proves with 100% certainty with a medical doctor’s definitive diagnosis that the health problem was directly caused by mold in the rental premises, and that the manager and/or owner was negligent.

This is why its important to conduct your business correctly and make sure your insurance is in order (along with  any vendors who perform services for you) because a good operator does not do things where insurance coverage is denied. 

But I would just offer an egregious lease renewal, and if they deny it then great.

I have seen some lenders get all the way into underwriting and deny the loan due to the "risk" of the collateral versus total vested interest.

(I do not know if she has a legit letter or not) Regardless, I understand FHA I can't deny her application based soley on the ESA, but if the one of the current tenant have allergy issue, can I deny then?

If they deny the rent, they will send you a letter asking if you will accept or deny the reduced amount.

Denis,Yes, but keep in mind some lenders have specific overlays and will not allow. 

Deny it and, well, prepare for a rather miserable existence of stress, anger, frustration repeating. 

Those photos (if you have them) should also be sent to the tenant and used if you go to court otherwise it is your word against theirs (should they deny leaving the mess).