Do black holes violate the first law of thermodynamics? When a black hole absorbs matter is it destroying that mass, thereby destroying energy, therefore violating the first law of thermodynamics?
 A: 
Do black holes violate the first law of thermodynamics?

No. See Wikipedia re the first law of thermodynamics: "The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic systems. The law of conservation of energy states that the total energy of an isolated system is constant; energy can be transformed from one form to another, but cannot be created or destroyed". If you throw a one-kilogram object into a zillion-kilogram black hole, the black hole mass increases by one kilogram. The object might get destroyed, but you can't destroy energy. Or create it. There are no perpetual motion machines. Energy is fundamental. Everything is made of it, including light and matter, and black holes.  

When a black hole absorbs matter is it destroying that mass, thereby destroying energy, therefore violating the first law of thermodynamics?

No. It destroys the matter, but the total mass stays the same, as does the total energy. In the scenario above, you start with one zillion and one kilograms, and you end up with with one zillion and one kilograms. The black hole's gravitational field increases a little because you increased its mass-energy by one kilogram.  
A: No, it does not. 
If it did, then it would not curve spacetime and it would not be a black hole.
A: Yes, a black hole can violate conservation of energy. According to JamelS' answer at Energy conservation and General Relativity, energy is not necessarily conserved in general relativity. According to my answer at Energy conservation and General Relativity, the electric field of a charged black hole can accelerate a charged particle but an electric field can't accelerate a charged black hole. If you have two charged black holes orbiting each other, they will accelerate each other only with their gravitational field and not with their electric field so total of their gravitational potential energy and the energy of their gravitational waves will remain constant but their electric potential energy will keep changing as they spiral into each other.
