$X$ is a closed compact Riemann surface. $\operatorname{Jac}(X)$ is the Jacobian of $X$. One can always embed $X$ into projective space by twisting line bundle over $X$ to an ample line bundle by raising high power enough.
$\operatorname{Jac}(X)$ is an abelian group. $\operatorname{Jac}(X)$ admits a projective embedding if it has a positive definite hermitian form/admits a divisor whose associated $\theta$ line bundle is globally generated.(This is basically saying the condition of admitting the projective embedding.)
For $X$ being elliptic curve and Riemann surface, $\operatorname{Jac}(X)\cong X$. Since all elliptic curve are non-singular for closed compact Riemann surface, I see both $X$ and $\operatorname{Jac}(X)$ can be projectivized and this is if and only if relationship.
$\textbf{Q1:}$ What is the relationship between $X$ projectivizable and $\operatorname{Jac}(X)$ projectivizable?(For closed compact Riemann surface $X$ is certainly projectivizable into $CP^3$.) I do not know whether this should also imply $\operatorname{Jac}(X)$ projectivizable. The book says there are some lattice which do not admit positive definite hermitian forms and hence it cannot be embedded into projective space.
$\textbf{Q2:}$ What is the relationship between meromorphic sections on $\operatorname{Jac}(X)$ and meromorphic sections on $X$?
Ref. Analytic Theory of Abelian Varieties by Swinnerton.