Cohomology of genus g surface fibration

Question

Let $F\to E\to X$ be a Serre fibration with connected base space $X$ and let the fibre $F$ be an orientable surface of genus $g$. Is there anything we can say about how $H^n(E)$ depends on $H^n(X)$? My ideas:

1. Maybe we should first assume $X$ to be simply connected and use Leray–Serre. The corresponding $E_2$-page of the spectral sequence should be of the form $$\begin{array}{ccc}\mathbb{Z}&H^1(X)&H^2(X)&\dotsb\\\mathbb{Z}^{2g} & \bigoplus^{2g}H^1(X)=0 & \bigoplus^{2g}H^2(X) & \dotsb\\\mathbb{Z} & H^1(X)=0 & H^2(X) & \dotsb\end{array}$$
2. Is there a way to generalize the Gysin-sequence, where $F=\mathbb{S}^2$, i. e. $g=0$?