Regarding your first question, I might know where your confusion is coming from. Some people also would refer to a Hecke algebra by the space of complex valued function $f$ on $SL_2(\mathbb{Z})$ or $GL_2^+(\mathbb{Z})$ biinvariant under $G$ respective $G'$ (the algebra multiplication defined via convolution), especially if you are coming from an adelic point of view and $G, G'$ are congruence. If $G$ is finite index in $G'$, there is a inclusion (by restriction) as well as a projection (by averaging over double cosets).
But the usual Hecke operators due to Hecke are really only a subalgebra of these algebras. But it is e.g. a commutative algebra if $G = SL_2(\mathbb{Z})$ or $G=GL_2^+(\mathbb{Z})$, because the $( GL_2(\mathbb{Q}_p), GL_2(\mathbb{Z}_p))$ is a Gelfand pair for each $p$.