The first question is answered in Serre's "[Propriétés galoisiennes ..." des points d'ordre fini des courbes elliptiques, Invent. Math. 15, 1972:4 (1972) 259--331], on page 304, section 5.2, exactly for this curve. In general this paper give a good way to determine for which $\ell$ the mod-$\ell$ representation is not surjective. Sage can do that efficiently for a given curve.
For every prime $p$ different from $\ell$ and $11$, the characteristic polynomial of $\rm{Frob}_p$ is indeed $T^2 - c_p T +p$ in $ \mathbb{F}{\ell}[T]$. \mathbb F_{\ell}[T]$. The isomorphism $\rm{Gal}(K {\ell}/\mathbb{Q}) _{\ell}/\mathbb Q) \to \rm{Aut}(E[\ell])=\rm{GL} 2(\mathbb{F}{\ell})$ _2(\mathbb F _{\ell})$ sends $\rm{Frob}_p$ to the Frobenius endomorphism $\phi:E[\ell] \to E[\ell]$ on $E/\mathbb{F}_p$. Your $c_p$ is the trace of $ \phi$ \phi$ and $p$ is the determinant of it since the Eichler-Shimura relatoin Eichler--Shimura relation shows that $c_p$ is the Fourier coefficent of the associated modular form. See this answer for why it is so.
