Songs of the Earth by Elspeth Cooper.
The protagonist is Gair. When we meet him he has just discovered he has magic powers, which is a death sentence since witches are burned at the stake. He is caught, tried and condemned, but his sentence is commuted to banishment. He is found by Alderan and taken to the school of magic in the town Pencruik on the Western Isles, where amongst other things he learns so shapeshift. Gair can shift into many different animals and these include a fire-eagle and a wolf as you mention.
The veil is the boundary between two worlds, and the first evidence it is thinning is when a stag being hunted tears through the veil between the worlds. I'm afraid I have forgotten what the other world is called and cannot find it by skimming the book - I think it's just referred to as The Hidden Kingdom.
The hellhound comes from the other world through a tear in the veil. It's referred to as Maegern's Hound in the book.