Proof that the push-forward of a sheaf is unstable

Question

Let $\pi:X\to C$ be a (proper) elliptic surface with $F$ as a general fiber and H a polarization on $X$.

Let $\mathcal F$ be a bundle on $C$ and let $\mathcal E:=\pi^* \mathcal F$.

Assume $\mathcal E$ unstable and let $\mathcal E_{max}$ be a maximal destabilizing bundle such that $\mathcal E_{max}\cong \pi^*\pi_*\mathcal E_{max}$.

Claim: $\mu(\pi_*\mathcal E_{max}) = \frac{c_1(\mathcal E_{max}).H / F.H}{\operatorname{rank} \mathcal E_{max} } > \mu(\mathcal F) = \frac{c_1(\mathcal F).H / F.H}{\operatorname{rank} \mathcal F}$, hence $\pi_*\mathcal E_{max}$ destabilizes $\mathcal F$.

I am trying to understand the above inequality: using the fact that $\mu (\mathcal E_{max}) > \mu(\mathcal E)$, we get

$\mu (\mathcal E_{max}) = \frac{c_1(\pi^*\pi_*\mathcal E_{max}).H}{\operatorname{rank} \mathcal E_{max} }> \mu(\pi^*\mathcal F) = \frac{c_1(\pi^*\mathcal F).H}{\operatorname{rank} \mathcal F }$.

Here I have a few questions:

1. Is $c_1(\pi^*\pi_*\mathcal E_{max}).H = c_1(\pi_*\mathcal E_{max}).H / F.H$? I tried using properties of chern classes like compatibility with pullback, projection formula, proper push-forward, but I couldn't make sense of it. Where does division by $F.H$ come from?
2. How to understand $c_1(\mathcal F).H$ in the right hand side where we intersect a divisor on $X$ with a divisor on $C$?

Thank you.