Hey I remember a book and I'm hoping you have read it and know what I'm talking about .

Think it had a snowy cave with a gold dragon and some adventurers on the cover (could be way off)
List of things I know



About a boy with a gray patch of hair who is in fact a wizard but does not know this at the start
He is the son I think of the villain of the story but does not know this at least at the beginning
At one point the go to a castle I think the king of that castle is a lion with a magical library that has every book ever in it but the book he needs is missing
The world they are in seems to be divided into sections all run by dragons as the leaders and the different sections have different color dragons
Dragon births are rare and a cycle of egg hatching is coming
The main character has a friend a magical black horse
Eventually a female sorceress joins his group, he saves her from harpies , she is frozen in a crystal I think in a house in the woods
At some point they go to a cave I think that is defended by armadillo people guarding something he needs to save the day from the evil wizard

This almost sounds like some Piers Anthony books I've read. Try looking at some of his work and see if you can find it there.