(Disclaimer: I am a mathematician, not a physicist, and I hated the one thermodynamics course I took at university several decades ago. If anything I write here looks wrong, it probably is.)

Let's try to do this from lower-level first principles.

The temperature is proportional to the average kinetic energy of the molecules.

Suppose you have particles with (mass,velocity) $(m_1,v_1)$ and $(m_2,v_2)$ and they collide perfectly elastically (which I believe is basically what happens for collisions between individual molecules). The centre-of-mass frame is moving with velocity $\frac{m_1{v}_1+m_2{v}_2}{m_1+m_2}$ relative to the rest frame, and there the velocities are $\frac{m_2}{m_1+m_2}(v_1-v_2)$ and $\frac{m_1}{m_1+m+2}(v_2-v_1)$. In this frame, the collision simply negates both velocities (since this preserves the net momentum of zero and the net kinetic energy) so now they are $\frac{m_2}{m_1+m_2}(v_2-v_1)$ and $\frac{m_1}{m_1+m_2}(v_1-v_2)$, so back in the rest frame the velocities are $\frac{(m_1-m_2)v_1+2m_2{v}_2}{m_1+m_2}$ and $\frac{(m_2-m_1)v2+2m_1{v}_1}{m_1+m_2}$. So the first particle's kinetic energy has changed by (scribble scribble) $2\frac{m_1}{(m_1+m_2{)}^2}[m_1(m_1-m_2)v_1{v}_2+2m_2^2{v}_2^2-4m_1{m}_2{v}_1^2]$ .

(I should say explicitly that these velocities are vectors and when I multiply two of them I mean the scalar product.)

Now, the particles with which any given particle of our cooling object comes into contact will have randomly varying velocities. I think the following may be bogus but let's suppose that the direction they're moving in is uniformly random, so the distribution of v2 is spherically symmetrical. (It may be bogus because it seems like two particles are more likely to collide when their velocities are opposed than when they are in the same direction, and if that's right then it will induce a bias in the distribution of v2.) In this case, the v1 v2 term averages out to zero and for any choice of |v2| we are left with constant . (KE2 - KE1). So as long as (1) our possibly-bogus assumption holds and (2) the rate at which object-particles and environment-particles interact isn't changing, the rate of kinetic energy change should be proportional to the average value of KE2-KE1, which is to say proportional to the difference in termperatures.

This is the (unmodified) Newton cooling law.

So if my calculations are right then any wrongness in the Newton law is a consequence of assumptions 1 and 2 above failing in whatever ways they fail.

I think assumption 2 is OK provided we keep the environment at constant temperature. (Imagine an environment-molecule somewhere near the surface of our object. If it's heading towards our object, then how long it takes before it collides with an object-molecule depends on how many object-molecules there are around, but not on how they're moving.)

I am suspicious of this "Lambert's law". Suppose the environment is at absolute zero -- nothing is moving at all. Then "Lambert's law" says that the rate of cooling should be infinite: our object should itself instantly drop to absolute zero once placed in an absolute-zero environment. Can that be right?

$\frac{dQ}{dt}\propto \frac{1}{T}-\frac{1}{T_{\infty }}$.

How's this square up the the Stefan Bolztmann law? I.e., for a black body, the power radiated is proportional to $T^4$. So if you had two black bodies radiating all their energy at each other, with one acting as a resevoir, you'd get a net outward flux of power of the form $P\alpha (T^4-T_{env}^4)$, right?

I'm going to open up with a technical point: it is important, not only in general but particularly in thermodynamics, to specify what quantities are being held fixed when taking partial derivatives. For example, you use this relation early on:

$\frac{dS}{dQ}=\frac{1}{T}$.

This is a relationship at constant volume. Specifically, the somewhat standard notation would be

${\frac{\partial U}{\partial S}|}_V=T$,

where U is the internal energy.  The change in internal energy at constant volume is equal to the heat transfer, so it reduces to the relationship you used.

That brings us to the lemma you wanted to use:

$\frac{dX}{dt}\propto \frac{dS}{dX}$.

To get what you wanted, it has to actually be the derivative with constant volume on the right, but then there's a problem: it doesn't succeed in giving you the time derivative of V since ${\frac{\partial S}{\partial V}|}_V=0$. 

Let's assume that problem with the lemma can somehow be fixed, though, for sake of discussion. There's another issue, which is that if the proportionality depends on thermodynamic variables, then you can have basically any relationship. For example, your heat equation:

$\frac{d{Q}_1}{dt}\propto \frac{d{S}_1}{d{Q}_1}-\frac{d{S}_2}{d{Q}_2}=\frac{1}{T_1}-\frac{1}{T_2}$.

If these proportionalities were $T_1^2$ and $T_2^2$, it would actually give Newton's law of cooling. For an ideal gas, the equation of state $U=\frac{3}{2}NkT$ means that the change in internal energy (which is just the heat transfer at constant volume) ought to be directly proportional to the temperature change, with no dependence on other thermodynamic variables (besides $N$) in the proportionality.

Now we have your formula for the derivative of temperature:

$\frac{dT}{dt}\propto -\frac{d^{2}S}{d{Q}^2}{T}^2(\frac{1}{T}-\frac{1}{T_{\infty }})$.

As a side note, I'm not sure how this is a heat capacity; it doesn't match any of the heat capacity formulas I remember. But the appearance of $T^2$ is notable; it makes it look a lot closer to Newton's law of cooling, and comparing it to the earlier equation for heat shows how the proposed proportionalities from the first lemma contain dependence on other thermodynamic variables. But you changed from $T_1$  and $T_2$  to $T$ and $T_{\infty }$ before this, so it's worth remembering that there should be a symmetric relationship between the two subsystems. Multiplying both of the inverse temperature terms by a single temperature produces an asymmetry in the time derivatives for the two subsystems.

This asymmetry in the temperature dependence would predict that one subsystem will heat faster than the other subsystem cools, which would tend to violate energy conservation. If we just imagine an ideal gas in two separate with an identical number of particles in each container, any temperature increase in one gas has to be exactly compensated by an identical magnitude temperature decrease in the other gas, since the internal energy is just proportional to temperature.

So I argue that this proposed law does not hold up.