= Solution
Use the source convention $Q=T_3+Y$, with electric charge measured in units of the positron charge. Expanding the <covariant derivative> in the <fermion> <kinetic term> gives interaction $-\bar f\gamma_\mu(gT_3A_3^\mu+g'YB^\mu)f$. Substituting the given neutral-field rotation, the coefficient of $Z^\mu$ is
$$
gc_WT_3-g's_WY=\frac g{c_W}(T_3-s_W^2Q),\qquad s_W=\sin\theta_W,\quad c_W=\cos\theta_W.
$$
The photon coefficient is $gs_W(T_3+Y)=eQ$, with $e=gs_W=g'c_W$. For a <Dirac field>, $T_3$ acts only on its left-handed component. Thus
$$
T_3P_L-Qs_W^2=\frac12\bigl[(T_3-2Qs_W^2)-T_3\gamma^5\bigr].
$$
The <weak neutral current> interaction consequently has the requested form, with
$$
\boxed{c_v=T_3-2Qs_W^2,\qquad c_a=T_3.}
$$
The <electroweak representation and hypercharge table> gives the following values for one generation:
|| Type
|| $T_3$ on left
|| $Q$
|| $Y_L$
|| $Y_R$
|| $c_v$
|| $c_a$
| <Neutrino>
| $1/2$
| $0$
| $-1/2$
| absent
| $1/2$
| $1/2$
| Charged <lepton>
| $-1/2$
| $-1$
| $-1/2$
| $-1$
| $-1/2+2s_W^2$
| $-1/2$
| Up-type <quark>
| $1/2$
| $2/3$
| $1/6$
| $2/3$
| $1/2-4s_W^2/3$
| $1/2$
| Down-type <quark>
| $-1/2$
| $-1/3$
| $1/6$
| $-1/3$
| $-1/2+2s_W^2/3$
| $-1/2$
These <neutral-current vector and axial couplings> use the source's vertex prefactor $g/(2c_W)$. They are twice the coefficients used with prefactor $g/c_W$; the latter convention must not be mixed into this calculation. For the neutrino, the <chiral projector> removes the inactive right-handed component automatically.
Set $G=g/(2c_W)$ and suppress the species label. For outgoing momenta $k_1,k_2$, the amplitude is, up to an irrelevant phase,
$$
\mathcal M=G\epsilon_\mu(p)\bar u(k_1)\gamma^\mu(c_v-c_a\gamma^5)v(k_2),\qquad p=k_1+k_2.
$$
The final-state <spin sum> is the <Dirac trace> of $\not k_1\gamma^\mu(c_v-c_a\gamma^5)\not k_2\gamma^\nu(c_v-c_a\gamma^5)$. Anticommutation of $\gamma^5$ with the <gamma matrices> makes its symmetric part
$$
T_{\mathrm{sym}}^{\mu\nu}=4(c_v^2+c_a^2)(k_1^\mu k_2^\nu+k_1^\nu k_2^\mu-g^{\mu\nu}k_1\cdot k_2).
$$
The vector-axial cross term is proportional to the antisymmetric <Levi-Civita symbol>, so it contracts to zero with the symmetric <massive vector polarization sum> $-g_{\mu\nu}+p_\mu p_\nu/M_Z^2$. Since $k_1^2=k_2^2=0$, the symmetric tensor is transverse to $p$ as well. Only $-g_{\mu\nu}$ remains, and
$$
-g_{\mu\nu}T_{\mathrm{sym}}^{\mu\nu}=8(c_v^2+c_a^2)k_1\cdot k_2=4M_Z^2(c_v^2+c_a^2).
$$
Multiplying by $G^2$ proves the <massless fermion Z decay spin sum>:
$$
\boxed{\sum_{\lambda,s_1,s_2}|\mathcal M|^2=\frac{g^2M_Z^2}{c_W^2}(c_v^2+c_a^2).}
$$
For the <decay width>, average over the three initial polarizations. Rotational invariance makes the integrated width the same for each initial spin state, so the <massive-vector spin average> is legitimate even for a single prepared polarization. In the rest frame, the delta functions in <two-body Lorentz-invariant phase space> leave
$$
d\Phi_2=\frac{d\Omega}{32\pi^2},\qquad\int d\Phi_2=\frac1{8\pi}.
$$
This follows by setting $\mathbf k_2=-\mathbf k_1$ and integrating $\delta(M_Z-2|\mathbf k_1|)$; the radial integration supplies a factor $1/2$. Hence
$$
\Gamma_f=\frac1{2M_Z}\frac13\sum|\mathcal M|^2\frac1{8\pi}=\boxed{\frac{g^2M_Z}{48\pi c_W^2}(c_v^2+c_a^2)}.
$$
For a <quark> species multiply by three colors, with strong-interaction radiative corrections when comparing with data. For one effectively massless active <neutrino>, $c_v=c_a=1/2$, so $\Gamma_\nu=g^2M_Z/(96\pi c_W^2)=G_FM_Z^3/(12\sqrt2\pi)$.
The total resonance width contains visible charged-lepton and hadron decays plus $N_\nu\Gamma_\nu$. Thus the measured invisible contribution determines
$$
\boxed{N_\nu=\frac{\Gamma_Z-\Gamma_{\mathrm{visible}}}{\Gamma_\nu}.}
$$
Early resonance measurements supported three light active <neutrino> species, with experimental uncertainties; the https://inspirehep.net/files/afa87f449c4b3cb87894b53955b564da[OPAL analysis of its 1989 data] explicitly compares the three- and four-species predictions. Under the question's assumption that every possible family has a massless <neutrino> with ordinary weak couplings, this constrains the number of such families. The inference does not exclude additional neutrinos heavier than $M_Z/2$ or <gauge singlet> sterile states with no ordinary $Z$ coupling. This is <light-neutrino counting from the Z width>.
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