My initial thought was to wonder whether you were just mis-remembering Seanan McGuire's Incryptid series, which definitely hits some of the points on your list (while totally missing others)! But a little more digging suggests that actually it's not them at all, and what you're remembering is probably the Bentwhistle the Dragon series by Paul Cude:

https://www.goodreads.com/author/show/5368048.Paul_Cude