Is every countable Dedekind domain the ring of integers of some number field? I tried googling different keywords, but did not find anything. Does anyone know of research in this area?

Nope. $\mathbb F_p[x]$, $\mathbb Q[x]$, and all other affine algebraic curves over countable fields, are countable Dedekind domains. None are the ring of integers of a number field. 

