= Solution
Write $\Gamma=SL_2(\mathbb Z)$ and use the <determinant-normalized slash operator>
$$
(f|_k\alpha)(\tau)=(\det\alpha)^{k/2}(c\tau+d)^{-k}f(\alpha\tau),\qquad
\alpha=\begin{pmatrix}a&b\\c&d\end{pmatrix},\quad\det\alpha>0.
$$
The <automorphy factor> identity proves $(f|_k\alpha)|_k\beta=f|_k(\alpha\beta)$, and positive scalar matrices act trivially. Let $\mathcal M_n$ be all integral matrices of determinant $n>0$. It is stable under left and right multiplication by $\Gamma$. Its left orbits have unique representatives
$$
\Pi_n=\left\{\begin{pmatrix}a&b\\0&d\end{pmatrix}:ad=n,\ a,d>0,\ 0\leq b<d\right\}.
$$
Indeed <Bézout's identity> reduces the first column to $(a,0)$, the determinant fixes $d=n/a$, and a row shear reduces $b$ modulo $d$. These are the <determinant-n matrix representatives for Hecke operators>. For composite $n$, $\mathcal M_n$ can contain more than one double coset; keeping every integral matrix, including nonprimitive ones, is essential.
Define the <Hecke operator> by
$$
\boxed{T_nf=n^{k/2-1}\sum_{\alpha\in\Pi_n}f|_k\alpha
=n^{k-1}\sum_{ad=n}d^{-k}\sum_{b=0}^{d-1}f\left(\frac{a\tau+b}{d}\right).}
$$
Changing a representative on the left leaves its summand unchanged. Right multiplication by $\Gamma$ permutes left orbits of $\mathcal M_n$, so the result satisfies the weight-$k$ transformation law. Each summand is holomorphic. Its expansion at infinity is bounded, and vanishes there when $f$ is a <cusp form>; the group has just one cusp class. Therefore $T_n$ preserves both $M_k$ and $S_k$, and $T_1$ is the identity.
For a <Fourier expansion> $f=\sum_{r\geq0}a_rq^r$, the sum over $b$ vanishes unless $d\mid r$. Substitute $r=dt$ and then collect the coefficient of $q^m$ to obtain the <Fourier coefficients of a composite-index Hecke operator>:
$$
\boxed{a_m(T_nf)=\sum_{h\mid\gcd(m,n)}h^{k-1}a_{mn/h^2}(f).}
$$
Here $\gcd(0,n)=n$, so the constant coefficient is $\sigma_{k-1}(n)a_0$; at $m=1$ it is $a_n$. This proves preservation of integral coefficients. At a prime,
$$
T_pf=p^{k-1}f(p\tau)+\frac1p\sum_{b=0}^{p-1}f((\tau+b)/p),\qquad
a_m(T_pf)=a_{pm}+p^{k-1}a_{m/p},
$$
with $a_{m/p}=0$ if $p\nmid m$.
We can prove the entire <Hecke multiplication relations> directly, not just quote them. On <formal power series> put $U_p(\sum a_mq^m)=\sum a_{pm}q^m$ and $V_p f(q)=f(q^p)$. Although these individual operators need not preserve level one, $U_pV_p=1$ and the coefficient formula gives
$$
T_{p^r}=\sum_{j=0}^r p^{j(k-1)}V_p^jU_p^{r-j}.
$$
Multiply on the left by $T_p=U_p+p^{k-1}V_p$. In the first part the $j=0$ term is $U_p^{r+1}$; every $j\geq1$ reduces by $U_pV_p=1$ to $p^{j(k-1)}V_p^{j-1}U_p^{r-j}$. The second part, together with the $j=0$ term, is $T_{p^{r+1}}$. The remaining sum is $p^{k-1}T_{p^{r-1}}$. Thus
$$
T_pT_{p^r}=T_{p^{r+1}}+p^{k-1}T_{p^{r-1}}\quad(r\geq1).
$$
For coprime indices, the coefficient divisors split independently in the displayed formula, proving $T_mT_n=T_{mn}$ when $(m,n)=1$. Induction with the prime-power recurrence gives $T_{p^r}T_{p^t}=\sum_{j=0}^{\min(r,t)}p^{j(k-1)}T_{p^{r+t-2j}}$. Combining the primes proves
$$
\boxed{T_mT_n=\sum_{h\mid\gcd(m,n)}h^{k-1}T_{mn/h^2}.}
$$
In particular the <modular Hecke algebra> is commutative and is generated by the prime operators.
For <cusp forms> the <Petersson inner product> is
$$
\langle f,g\rangle=\int_{\Gamma\backslash\mathbb H}f(\tau)\overline{g(\tau)}y^k\frac{dx\,dy}{y^2}.
$$
Cusp decay makes it finite and positive definite. The <Hecke operators are self-adjoint for the Petersson inner product>. To see the change-of-variables mechanism, fix a double coset $\Gamma\alpha\Gamma\subset\mathcal M_n$ and put $H=\Gamma\cap\alpha^{-1}\Gamma\alpha$. Representatives $\gamma$ of $H\backslash\Gamma$ give left representatives $\alpha\gamma$. Unfolding the sum in $\langle\sum_\gamma f|_k\alpha\gamma,g\rangle$ changes the integral over $\Gamma\backslash\mathbb H$ to the integral over $H\backslash\mathbb H$ of $(f|_k\alpha)\overline g$. The substitution $w=\alpha\tau$ preserves $dx\,dy/y^2$ and has $\Im w=(\det\alpha)y/|c\tau+d|^2$; these factors move the slash from $f$ to $g|_k\alpha^{-1}$. Refold over $\alpha H\alpha^{-1}=\Gamma\cap\alpha\Gamma\alpha^{-1}$ to get the reverse double coset. Since scalar matrices act trivially, $\alpha^{-1}$ has the same slash action as its adjugate $\alpha^*=n\alpha^{-1}$, an integral matrix of determinant $n$. The adjugate involution permutes all the double cosets in $\mathcal M_n$. The common real factor $n^{k/2-1}$ is unchanged, giving $\langle T_nf,g\rangle=\langle f,T_ng\rangle$.
Since $S_k$ is finite dimensional, commuting self-adjoint <linear operators> have a common <orthonormal basis> of <Hecke eigenforms>: diagonalize one operator, restrict the others to its eigenspaces, and repeat until these spaces cannot split further. If $f$ is a nonzero common <Hecke eigenform> with $T_nf=\lambda_nf$, the coefficient at $q$ gives $a_n=\lambda_na_1$. Were $a_1=0$, every positive coefficient would vanish, contrary to $f\ne0$ being cuspidal. Normalize $a_1=1$; then $\lambda_n=a_n$, and the <Hecke multiplication relations> yield
$$
a_ma_n=\sum_{h\mid\gcd(m,n)}h^{k-1}a_{mn/h^2},\qquad
a_{p^{r+1}}=a_pa_{p^r}-p^{k-1}a_{p^{r-1}}.
$$
The <eigenvalues> are real by self-adjointness. Their multiplicativity and prime-power recurrence give the <Euler product of a Hecke eigenform>, initially in an absolutely convergent half-plane:
$$
\boxed{L(f,s)=\prod_p\left(1-a_pp^{-s}+p^{k-1-2s}\right)^{-1}.}
$$
The <Mellin transform> in Question 5 supplies its analytic continuation and functional equation.
Finally, for positive even $k\geq4$, $M_k=\mathbb C E_k\oplus S_k$, by subtracting the constant coefficient times the normalized <Eisenstein series>. The divisor identity
$$
\sigma_{k-1}(m)\sigma_{k-1}(n)=\sum_{h\mid\gcd(m,n)}h^{k-1}\sigma_{k-1}(mn/h^2)
$$
follows by multiplying the geometric sums at each prime; it and the constant-coefficient formula prove $T_nE_k=\sigma_{k-1}(n)E_k$. Thus $M_k$ has an <Eisenstein series> eigenline and a basis of cuspidal <Hecke eigenforms>. Odd and negative weights have no forms, $M_2=0$, and on $M_0=\mathbb C$ the operator is the scalar $\sigma_{-1}(n)$. These statements fix the normalization and describe the exceptional small weights as well.
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