You can only claim a tax deduction for charitable contributions made to a 501c3 organization. So far Pilots N Paws does not have that yet, although I understand it is in the works. However, many of the rescues probably are.
Angel Flight missions are tax deductible. We get a report at the end of the year on our donations. I think that is required by tax regulations. Work with the specific rescues you fly for to ensure you have the proper documentation. Contact your tax professional for advice.
As far as the FAA regulations, I believe at one time the FAA tried to cite a pilot for taking a tax deduction for a charitable flight, but the action was overturned. That is a good question for AOPA (which is where I think I read it).