Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is false, already for the trivial bundle of rank one $E_1=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any non-empty set $U \subset \mathbb C$ it is impossible to approximate the $C^{\infty}$ function $\bar z$ by holomorphic functions since the limit of a sequence of holomorphic functions on $U$ is a holomorphic function on $U$. [The limit is to be understood in the sense of uniform convergence on compact subsets of $U$. There is also an $L^2$- version stating that $L^2(U)\cap \mathcal O(U)$ is a Hilbert subspace of $L^2(U)$, so that a sequence of holomorphic functions converging only in the $L^2$ sense nevertheless has a holomorphic limit. ]
For your "another question", the answer is also "no":
Take $E_2=X\times \mathbb C^2$ , the trivial rank-2 bundle. Then you cannot write the section $(1,\bar z)\in C^{\infty}( U,E_2)$ as $(1,\bar z)=\phi (z) (f(z),g(z))$ with $f,g$ holomorphic and $\phi$ smooth , since $\phi g/\phi f=g/f$ is meromorphic while $\bar z/1=\bar z$ is not ( notice that $\phi$ never vanishes since $1=\phi f$)