My question is: Is the homology of (C∗,∂) isomorphic to the singular homology of E?

Yes. Observe that the singular complex functor sends Serre fibrations to Kan fibrations. Thus, the map $$\def\Sing{\mathop{\rm Sing}} \Sing p\colon \Sing E\to \Sing B$$ is a Kan fibration.

Denote by $T$ the underlying simplicial set of the triangulation of $B$ so that we have a homeomorphism $|T|→B$ and an adjoint simplicial weak equivalence $t\colon T→\Sing B$.

Now the chain complex $C$ is simply the simplicial chains of the pullback of the span of simplicial sets $$T→\Sing B←\Sing E.$$

The base change of $\Sing p$ along $t$ yields a map $P→T$.

One leg of the above span is a Kan fibration, therefore the pullback computes the homotopy pullback. The map $T→\Sing B$ is a weak equivalence, hence so is its base change $P\to \Sing E$.

Thus, the map $P→T$ is weakly equivalent to $\Sing p$, so the chain complex of $P$ is quasi-isomorphic to the chain complex of $\Sing E$. The chain complex of $P$ is precisely $C$.

My question is: Is the homology of $(C_*,\partial)$ isomorphic to the singular homology of E?

Yes. Observe that the singular complex functor sends Serre fibrations to Kan fibrations. Thus, the map $$\def\Sing{\mathop{\rm Sing}} \Sing p\colon \Sing E\to \Sing B$$ is a Kan fibration.

Denote by $T$ the underlying simplicial set of the triangulation of $B$ so that we have a homeomorphism $|T|→B$ and an adjoint simplicial weak equivalence $t\colon T→\Sing B$.

Now the chain complex $C$ is simply the simplicial chains of the pullback of the span of simplicial sets $$T→\Sing B←\Sing E.$$

The base change of $\Sing p$ along $t$ yields a map $P→T$.

One leg of the above span is a Kan fibration, therefore the pullback computes the homotopy pullback. The map $T→\Sing B$ is a weak equivalence, hence so is its base change $P\to \Sing E$.

Thus, the map $P→T$ is weakly equivalent to $\Sing p$, so the chain complex of $P$ is quasi-isomorphic to the chain complex of $\Sing E$. The chain complex of $P$ is precisely $C$.