= Solution
For the two unlabelled follow-up requests, distinguish re-equilibrating <effusion> from <collisionless effusion>. The calculations below show that escaping atoms carry mean <kinetic energy> $2k_BT$, whereas a bulk atom has mean <kinetic energy> $3k_BT/2$. If the aperture is small and interatomic collisions maintain near-equilibrium conditions, faster atoms escape preferentially, and the remaining gas relaxes toward a lower-temperature <Maxwell-Boltzmann distribution>. No <heat> supplied by the walls is available to replace the lost energy.
For a monatomic gas at fixed container volume, <conservation of energy> gives a quantitative check on this <effusive cooling> picture:
$$
d\left(\frac32Nk_BT\right)=2k_BT\,dN,\qquad \boxed{\frac{dT}{T}=\frac13\frac{dN}{N}.}
$$
Since $dN<0$, the temperature falls. Under these idealizations $T\propto N^{1/3}$; the gas is not left with a permanently truncated velocity tail when collisions rapidly restore equilibrium.
If interatomic collisions are absent, the individual surviving atoms do not rethermalize to a new common temperature. Their <speed distribution> becomes selectively depleted, generally favouring slower survivors, and their angular and spatial distribution can also develop structure determined by which trajectories reach the hole. A single lower <temperature> does not describe the full nonequilibrium distribution. With a direction-randomizing but energy-preserving wall model, an illustrative survival law is $f(v,t)\propto f(v,0)e^{-\kappa vt}$; specular walls can give more detailed directional selection and even persistent nonescaping trajectories. The question supplies no wall-scattering or container-geometry model from which a unique collisionless loss law could be inferred.
For the labelled wall integrals, choose the polar axis toward the wall so that incoming atoms have $0\leq\theta\leq\pi/2$, and put
$$
\alpha=\frac{m}{2k_BT},\qquad A=\left(\frac{\alpha}{\pi}\right)^{3/2},\qquad I_j=\int_0^\infty v^j e^{-\alpha v^2}\,dv.
$$
The needed radial integrals are $I_3=1/(2\alpha^2)$, $I_4=3\sqrt\pi/(8\alpha^{5/2})$ and $I_5=1/\alpha^3$. They are related to the supplied integrals by rescaling the speed.
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