Dear Colin , for $X$  a holomorphic connected manifold, denote by $\mathcal M (X)$ its field of meromorphic functions.

A) It is not true that a germ of holomorphic function $f_x\in \mathcal O_{X,x}$ is induced by a global meromorphic function : many compact complex manifolds only have $\mathbb C$ as meromorphic functions:
$\mathcal M (X)=\mathbb C$. There is an example with $X$ a surface in Shafarevich's Basic Algebraic Geometry, volume 2, page 164.

B) The best analogon to Theorem B is probably Cartan-Serre's result that for any coherent sheaf $\mathcal F$ on the compact manifold $X$, the cohomology vector spaces $H^q(X,\mathcal F), q\geq 1$ are finite-dimensional  over $\mathbb C$.

(Original article:  Cartan-Serre, C.R.Acad.Sci. Paris **237** (1953), 128-130)