I want to inquire if there exists a Kunneth formula of underlined form:
Let $X$ and $Y$ be two compact complex manifolds with two holomorphic vector bundles $E$ and $F$ on $X$ and $Y$ respectively, define $\pi_1\colon X\times Y\rightarrow X$ as the projection onto $X$ and $\pi_2\colon X\times Y\rightarrow Y$ as the projection onto $Y$. Then we have \begin{align*} H^q(X\times Y,\pi_1^*E\otimes\pi_2^*F)=\bigoplus_{i+j=q}H^i(X,E)\otimes H^j(Y,F) \end{align*} Here all the sheafs are sheafs of holomorphic sections of the corresponding vector bundles. If it's not true in the above form, does anything of this sort exist, may be with some more asumptions?
Yes. See here on Mathoverflow or here on the Stacks project.