= Solution
The prescribed photon budget requires a collapsed mass fraction $f_{\rm coll}=3/3000=10^{-3}$, assuming baryons trace the collapsed mass fraction and using the supplied effective photon yield. No extra photon-escape or recombination correction should be counted on top of this stipulated budget.
For spectral slope $n=-2$, $q=1/6$. The mass-scale measurement gives
$$
\sigma(M_{\min},3)=0.5\left(\frac{64M_{\min}}{M_{\min}}\right)^{1/6}=1.
$$
Both relevant epochs are matter-dominated, so the <linear growth factor> ratio is $D(z)/D(3)=4/(1+z)$, independent of any present-day normalization convention. Thus $\sigma(M_{\min},z)=4/(1+z)$. The <Press-Schechter formalism> gives
$$
\operatorname{erfc}\left[\frac{\delta_{\rm sc}(1+z)}{4\sqrt2}\right]=10^{-3}.
$$
Using the supplied inverse value,
$$
\boxed{1+z_{\rm reion}=\frac{4\sqrt2\,(2.33)}{1.686}\simeq7.82,\qquad z_{\rm reion}\simeq6.8.}
$$
This is the <photon-budget reionization threshold> in the specified toy model, not a measurement of the full astrophysical reionization history.
To estimate the characteristic mass, use the non-dissipative <spherical-collapse model> result $\Delta_{\rm vir}=18\pi^2$. At this high redshift the background matter density is practically the <critical density>. Circular <virial velocity> then obeys
$$
v_{\rm vir}^2=\frac{GM}{r_{\rm vir}},\qquad
M=\frac{4\pi}{3}r_{\rm vir}^3\Delta_{\rm vir}\frac{3H^2}{8\pi G},
\qquad v_{\rm vir}=3\pi H r_{\rm vir}.
$$
Consequently
$$
M_{\min}=\frac{v_{\rm vir}^3}{3\pi GH(z)},\qquad
H(z)\simeq72\sqrt{0.25}(1+z)^{3/2}\,\mathrm{km\,s^{-1}\,Mpc^{-1}}.
$$
For $v_{\rm vir}=9\,\mathrm{km\,s^{-1}}$, this gives $H\simeq7.87\times10^2\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and \b[$M_{\min}\simeq2.3\times10^7M_\odot$], with physical virial radius about $1.21\,\mathrm{kpc}$.
At the threshold mass the <halo peak height> is $\nu=\sqrt2(2.33)\simeq3.30$, so
$$
n(M_{\min},z)M_{\min}=\frac{\bar\rho_{m,0}}{M_{\min}}F,\qquad
F=\sqrt{\frac2\pi}\frac16\nu e^{-\nu^2/2}\simeq1.92\times10^{-3}.
$$
The present mean matter density is $\bar\rho_{m,0}=2.775\times10^{11}\Omega_{\rm mat}h^2M_\odot\,\mathrm{Mpc}^{-3}\simeq3.60\times10^{10}M_\odot\,\mathrm{Mpc}^{-3}$. Hence the differential abundance per logarithmic mass interval is about $3.0\,\mathrm{Mpc}^{-3}$ and
$$
\boxed{\bar l=\bigl[n(M_{\min},z)M_{\min}\bigr]^{-1/3}
=\left(\frac{M_{\min}}{\bar\rho_{m,0}F}\right)^{1/3}\simeq0.69\,\mathrm{Mpc}\ \text{comoving}.}
$$
Rounded inputs and the supplied approximate factor justify quoting about $0.7\,\mathrm{Mpc}$. The corresponding physical separation is about $0.088\,\mathrm{Mpc}$ at this epoch. The requested <halo spacing from a differential mass function> uses a logarithmic mass bin of order unity; it is not obtained by identifying $f_{\rm coll}\bar\rho/M$ with that differential abundance. A cumulative number density above the threshold requires a separate mass integral.
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