The answer is no. Or at least not necessarily. At least for projectivity.

It is consistent that there is a vector space $V$ over $\Bbb F_2$ (or really any finite field) whose underlying set is amorphous, namely it cannot be partitioned into two infinite subsets.

This vector space is not finitely generated, but every proper subspace has a finite dimension.

Pick any such subspace $V'$ and consider $W=V/V'$. Then it is not very difficult to show that the following holds:

1. $W$ is a vector space whose underlying set is amorphous, and every proper subspace is finitely generated. But $|W|\neq|V|$.
2. There is no non-zero homomorphism from $W$ to $V$.
3. The natural quotient map is surjective from $V$ to $W$.

So $W$ is not projective. This also gives an example of an exact sequence which doesn't split.