Does there exist a complete, finitely axiomatizable, first-order theory $T$ with exactly 3 countable non-isomorphic models?
A few relevant comments:
There is a classical example of a complete theory with exacly $3$ models. This theory is not finitely axiomatizable (For the trivial reason that the language is infinite).
In this post, Javier Moreno explains how to rephrase this example in a finite language. Still, the theory is not finitely axiomatizable.
Some less relevant comments:
I would like to know if finite axiomability has ever be asked in this context.
There have been research in connection with stability. Lachlan has proved that a superstable theory with finitely many countable models is $\omega$-categorical. And it is still open if this can be extended to all stable theories.