Let $X$ be a smooth projective variety over a fixed field $k$ (take $k = \mathbb{C}$ if necessary). For a vector bundle $E$ on $X$, $ch(E)$ will be in the Chow ring.

$\textbf{Question 1: }$ If $\mathcal{E}$ is a locally free sheaf of rank $n$ on $X$, (with associated vector bundle $E$) can one recover the dimensions of the sheaf cohomology groups $\dim_k H^i(X, \mathcal{E})$ from the total chern class $\textrm{ch}(E)$? How about just $\dim_k H^0(X, \mathcal{E})$? If not that, how about if $E$ is just a line bundle? Can we at least determine if $\mathcal{E}$ has global sections?

$\textbf{Question 2: }$ In the case $k = \mathbb{C}$, can one recover the dimensions of the singular cohomology groups $\dim_k H^i_{sing} (X, k)$ from total chern classes of various bundles? We can recover the Euler characteristic of $X$ as $\int_X c_n(T_X)$. In the case of curves, we can even recover the geometric genus (since this is a degenerate case: the Euler characteristic and geometric genus encode the same information). Can we recover the geometric genus of $X$ if $\dim X > 1$ from chern classes of various bundles?

$\textbf{Question 3: }$ Is there a good example to indicate the kind of information that $\textrm{ch}(T_X)$ carries about $X$ beyond it's Euler characteristic?

$\textbf{Question 4: }$ Colloquially, people refer to the Chow ring as giving a "homology theory". In the case $k = \mathbb{C}$, can one recover the usual (singular) homology groups $H_i(X,\mathbb{Z})$ from the Chow groups? If not, what about $H_i(X, \mathbb{Q})$?