= Solution
For a <cochain complex> $E$, define the <shift of a cochain complex> by
$$
\boxed{(\Sigma E)^i=E^{i+1},\qquad d_{\Sigma E}^i=-d_E^{i+1}.}
$$
For a <cochain map> $a:E\to F$, put $(\Sigma a)^i=a^{i+1}$. This is again a <cochain map>, so the construction is a functor. The sign ensures the standard cone identities; the shifted differential squares to zero. Iteration gives $E[r]^i=E^{i+r}$ and differential $(-1)^rd_E$, and $H^i(E[r])=H^{i+r}(E)$.
For $\phi:E\to F$, its algebraic <mapping cone> is
$$
\boxed{C(\phi)^i=F^i\oplus E^{i+1},\qquad d_C(f,e)=(d_Ff+\phi e,-d_Ee).}
$$
Applying this differential twice gives first component $d_F^2f+d_F\phi e-\phi d_Ee=0$, since $\phi$ is a <cochain map>, and second component $d_E^2e=0$. The maps
$$
u:F\to C(\phi),\quad u(f)=(f,0),\qquad v:C(\phi)\to\Sigma E,\quad v(f,e)=e
$$
are <cochain maps>: $d_Cu(f)=(d_Ff,0)=u(d_Ff)$ and $v(d_C(f,e))=-d_Ee=d_{\Sigma E}v(f,e)$. They are maps of complexes of <sheaves>, because every displayed operation is a <sheaf> morphism in each degree.
The <bounded below derived category of sheaves> $D(X)=D^+(X)$ can be constructed by first quotienting the category of bounded below <cochain complexes> by <cochain homotopy> and then formally inverting all <quasi-isomorphisms>. A morphism may be represented by a roof $E\xleftarrow{\sim}E'\to F$. Alternatively, take <injective resolutions of sheaves> and use homotopy classes of maps between the resulting injective complexes. The <cohomology sheaf> functors descend to the localization, because they already identify homotopic maps and turn <quasi-isomorphisms> into isomorphisms. If $E\cong0$ in $D(X)$, applying these functors gives $\mathcal H^i(E)=0$ for every $i$. Conversely, if all these <sheaves> vanish, the unique map $E\to0$ is a <quasi-isomorphism>, which becomes invertible in $D(X)$. Hence
$$
\boxed{E\cong0\text{ in }D(X)\quad\Longleftrightarrow\quad\mathcal H^i(E)=0\text{ for every }i.}
$$
An <exact triangle in a derived category> is a triangle isomorphic to a cone triangle
$$
E\xrightarrow{\phi}F\xrightarrow{u}C(\phi)\xrightarrow{v}\Sigma E.
$$
In particular the last arrow is part of the structure; a mere sequence of three objects with successive composite zero does not suffice.
Suppose $0\to E\xrightarrow{\phi}F\xrightarrow{\psi}G\to0$ is degreewise exact. Define
$$
r:C(\phi)\to G,\qquad r(f,e)=\psi(f).
$$
Then $r(d_C(f,e))=\psi(d_Ff)+\psi\phi(e)=d_G\psi(f)=d_Gr(f,e)$, so $r$ is a <cochain map>. It is natural under maps of short exact sequences. It is degreewise surjective as a <sheaf> map, and its kernel can be identified with $E^i\oplus E^{i+1}$ using $(a,e)\mapsto(\phi(a),e)$. In these coordinates the kernel differential is
$$
d(a,e)=(d_Ea+e,-d_Ee).
$$
There is a <contracting homotopy> $h^i(a,e)=(0,a)$ in degree $i-1$:
$$
dh(a,e)=(a,-d_Ea),\qquad hd(a,e)=(0,d_Ea+e),\qquad dh+hd=\operatorname{id}.
$$
Thus the kernel has zero <cohomology sheaves>. The <cohomology-sheaf exact sequence> is the sequence of the short exact sequence of complexes $0\to\ker r\to C(\phi)\to G\to0$, or equivalently its ordinary complex-cohomology sequence at every stalk. It follows that $r$ induces isomorphisms of all <cohomology sheaves>, proving
$$
\boxed{C(\phi)\xrightarrow{\sim}G\text{ is a natural quasi-isomorphism}.}
$$
Under this isomorphism the cone triangle becomes the <exact triangle>
$$
E\xrightarrow{\phi}F\xrightarrow{\psi}G\xrightarrow{\delta}\Sigma E,
$$
where $\delta=v\circ r^{-1}$ in the derived category.
For the final construction write $U=S^2\setminus S^1$ and set $A=i_!\mathbb Q_U$, $B=\mathbb Q_{S^2}$, $C=j_*\mathbb Q_{S^1}$, each initially placed in degree zero. The stated sequence $0\to A\to B\to C\to0$ is exact on stalks: on $U$ its first map is an isomorphism and its last <sheaf> is zero; on the equator its first <sheaf> is zero and the second map is an isomorphism. Take its connecting morphism
$$
\boxed{\mu=\delta:C\longrightarrow A[1]\quad\text{in }D(S^2).}
$$
The source has only $\mathcal H^0(C)=C$, whereas the target has only $\mathcal H^{-1}(A[1])=A$. Therefore every induced map $\mathcal H^i(\mu)$ is zero: in any degree at least one of its source and target is zero. This is a <ghost morphism in a derived category>.
To prove the derived morphism itself is nonzero, apply derived global sections, giving the <hypercohomology> sequence. The <extension by zero> identifies $H^*(S^2;A)$ with the <compactly supported cohomology> of $U$, and closed direct image identifies $H^*(S^2;C)$ with $H^*(S^1;\mathbb Q)$. The complement $U$ is two open disks, each homeomorphic to $\mathbb R^2$, so
$$
H_c^q(U;\mathbb Q)=\begin{cases}\mathbb Q^2,&q=2,\\0,&q\ne2.\end{cases}
$$
The degree-one part of the resulting exact sequence is
$$
0=H^1(S^2;\mathbb Q)\longrightarrow H^1(S^1;\mathbb Q)=\mathbb Q\xrightarrow{\mu_*}H_c^2(U;\mathbb Q)=\mathbb Q^2\longrightarrow H^2(S^2;\mathbb Q)=\mathbb Q\longrightarrow0.
$$
Thus $\mu_*$ is injective, in particular nonzero. With the hemispheres oriented from $S^2$, the last map is $(a,b)\mapsto a+b$, so an appropriately oriented equator generator maps to $(1,-1)$. A zero morphism in the <derived category of sheaves> would induce a zero map after derived global sections and taking <cohomology>. Consequently \b[$\mu\ne0$, although $\mathcal H^i(\mu)=0$ for every $i$]. This establishes explicitly that the collection of cohomology-sheaf functors detects zero objects but is not faithful on morphisms.
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