About question 1. You can recover $\sum(-1)^i\dim H^i(X,E)$ by Riemann--Roch. But the individual cohomolgy groups cannot be recovered. For example, let $X$ be a curve of positive genus $g$ and $E$ a line bundle of degree $0$. If $E$ is generic it has $H^0 = H^1 = 0$, but for trivial bundle you have $\dim H^0 = 1$, $\dim H^1 = g$.
EDIT. Another example showing that the Chern classes with values in the Chow ring also don't help. Let $X = C \times P^1$ with $C$ being a curve of positive genus. Let $E = p^*L \oplus p^*L^{-1}$, where $L$ is a line bundle of degree zero on $C$ and $p:C \times P^1 \to P^1$ is the projection. Then $c_1(E) = 0$ and $c_2(E) = 0$ in the Chow ring. However, the dimension of the cohomology groups depend on $L$.
About question 1. You can recover $\sum(-1)^i\dim H^i(X,E)$ by Riemann--Roch. But the individual cohomolgy groups cannot be recovered. For example, let $X$ be a curve of positive genus $g$ and $E$ a line bundle of degree $0$. If $E$ is generic it has $H^0 = H^1 = 0$, but for trivial bundle you have $\dim H^0 = 1$, $\dim H^1 = g$.