= Solution
The <exterior derivative> is an $\mathbb R$-linear operator $d:\Omega^r(M)\to\Omega^{r+1}(M)$ which agrees with the <differential of a smooth function> on degree zero, satisfies $d^2=0$, and obeys the <graded Leibniz rule>
$$
d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^r\alpha\wedge d\beta,
\qquad \alpha\in\Omega^r(M).
$$
For the <uniqueness of the exterior derivative from its axioms>, first note that these axioms force locality. If $\alpha$ vanishes near $p$, choose a <smooth bump function> $\chi$ equal to one near $p$ and supported where $\alpha=0$. Then $0=d(\chi\alpha)=d\chi\wedge\alpha+\chi d\alpha$ gives $d\alpha(p)=0$. Thus $d\alpha(p)$ depends only on the form near $p$.
In a coordinate chart write $\alpha=\sum_I a_I\,dx^{i_1}\wedge\cdots\wedge dx^{i_r}$. Since $d(dx^i)=d^2x^i=0$, the axioms force
$$
d\alpha=\sum_{I,j}\partial_j a_I\,dx^j\wedge dx^{i_1}\wedge\cdots\wedge dx^{i_r}.
$$
This also proves existence. Define $d$ by that formula in each chart. The product rule for <partial derivatives> gives the <graded Leibniz rule>, and symmetry of mixed <partial derivatives> against antisymmetry of the <wedge product of differential forms> gives $d^2=0$. On overlaps it still agrees with the differential of every function, by the <chain rule>, so the preceding uniqueness argument in the second coordinate system makes the formulas agree. They therefore glue to a global <exterior derivative>.
For a <differential one-form> $\omega=\sum_i a_i dx^i$, the <exterior derivative of a one-form evaluated on vector fields> follows directly:
$$
d\omega(X,Y)=\sum_i\bigl(X(a_i)Y^i-Y(a_i)X^i\bigr).
$$
In $X(\omega(Y))-Y(\omega(X))$, the additional terms are $\sum_i a_i[X(Y^i)-Y(X^i)]=\omega([X,Y])$, where the bracket is the <Lie bracket of vector fields>. Hence
$$
\boxed{d\omega(X,Y)=X\bigl(\omega(Y)\bigr)-Y\bigl(\omega(X)\bigr)-\omega([X,Y])}.
$$
The <de Rham cohomology> is
$$
H^r_{\mathrm{dR}}(M)=
\frac{\ker(d:\Omega^r(M)\to\Omega^{r+1}(M))}
{\operatorname{im}(d:\Omega^{r-1}(M)\to\Omega^r(M))},
$$
with zero denominator in degree zero. The <Poincaré lemma> states that a <closed differential form> of positive degree on a star-shaped open subset of Euclidean space is an <exact differential form>; consequently every closed positive-degree form is locally exact on a <smooth manifold>.
For the <suspension isomorphism for top de Rham cohomology of spheres>, put $U=S^n\setminus\{\text{south pole}\}$ and $V=S^n\setminus\{\text{north pole}\}$. Both are diffeomorphic to $\mathbb R^n$, while $W=U\cap V$ deformation retracts onto the equatorial $S^{n-1}$. The <Mayer--Vietoris sequence for de Rham cohomology> comes from the <short exact sequence> of <cochain complexes>
$$
0\longrightarrow\Omega^*(S^n)
\longrightarrow\Omega^*(U)\oplus\Omega^*(V)
\xrightarrow{(\alpha,\beta)\mapsto\alpha|_W-\beta|_W}
\Omega^*(W)\longrightarrow0.
$$
The final map is surjective: a <partition of unity> $\chi_U+\chi_V=1$ lets a form $\eta$ on $W$ be lifted by extending $\chi_V\eta$ to $U$ and $-\chi_U\eta$ to $V$ by zero. The <Poincaré lemma> makes $H^{n-1}(U),H^{n-1}(V),H^n(U),H^n(V)$ vanish when $n>1$. Exactness gives
$$
0\longrightarrow H^{n-1}_{\mathrm{dR}}(W)
\xrightarrow{\delta}H^n_{\mathrm{dR}}(S^n)\longrightarrow0.
$$
If $j:S^{n-1}\hookrightarrow W$ is the equatorial inclusion, <homotopy invariance of de Rham cohomology> makes $j^*$ an <isomorphism>. Therefore
$$
\boxed{j^*\delta^{-1}:H^n_{\mathrm{dR}}(S^n)\longrightarrow H^{n-1}_{\mathrm{dR}}(S^{n-1})}
$$
is injective, indeed an <isomorphism>. Concretely, choose local primitives $d\alpha_U=\omega|_U$, $d\alpha_V=\omega|_V$, and send $[\omega]$ to $[j^*(\alpha_U-\alpha_V)]$. Their difference is closed; changing either primitive changes it by an exact form, so this agrees with the cohomological construction.
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