kingledion gave a great overview on the general scientific quality of the papers, but I don't think the title question has been really answered yet.
Is the Unified Theory of Climate (Nikolov & Zeller) compatible with the AGW/GHG theory in any respect?
Yes.
First, I too would comment on that preceding paper (On the average temperature of airless spherical bodies...). What it's concerned with is the equation
$$
T_\mathrm{e} = \left(\frac{S_0\cdot(1-\alpha_\mathrm{p})}{4\cdot\epsilon\cdot\sigma}\right)^{1/4},
$$
which derives from the Stefan-Boltzmann law the radiating equilibrium temperature $T_\mathrm{e}$. The main point of the paper is that this temperature is not in any way physically meaningful for airless bodies. Which is completely correct: airless bodies do indeed not have a “representative mean temperature”, because the integral over a function of some quantity (here, fourth power of temperature) isn't directly related to the function value of the average of the quantity†. For a body without atmosphere like the moon, temperature varies extremely strongly over the surface, and thermal emission is completely dominated by the hot regions.
This does not really give rise to any interesting conclusions though, because $T_\mathrm{e}$ isn't of much interest for climatology anyway. Any proper climate model is well aware of the importance of locally varying temperatures, as Volokin/ReLlez themselves admit:
It should be pointed out that global climate models intrinsically account for Hölder’s inequality by virtue of being three-dimensional and explicitly resolving the spatial heterogeneity of radiation absorption and emission (as well as other energy transport processes) within the context of a spherical geometry. However, 3-D models have not historically been applied to assess
the strength of Earth’s ATE (GE). Hence, our critique is strictly directed towards the effective emission-temperature formula (3) and other similar 1-D radiative-transfer models (e.g. Manabe and Möller 1961; Manabe and
Strickler 1964).
One could also add here that $T_\mathrm{e}$ is not quite so meaningless for a planet with substantial atmosphere but no greenhouse gases, because here the spatial temperature variations are less pronounced, yet the surface is in radiative equilibrium with the sun/space. I'll come back to that.
Now to the paper you're asking about. It does two things:
Fit generic models without a-priori physical motivation to observed values of pressure, density, greenhouse-gas content and observed and vacuum-predicted temperatures.
Try to take new physical conclusions from the fit result.
kingledion discussed well how sensible this approach is in principle. Yes, it is sensible to just fit the available data, if that's all you have available. It can even be valid to inter- or extrapolate from that data somewhat, using the fitted models as a guide. Nikolov/Zeller emphasize that they chose models which make such extrapolation particularly robust. Ok.
But there's one principle that anybody using statistical methods surely must always keep in mind: correlation does not imply causation. What Nikolov/Zeller “find out” is that you get a marvellous fit when considering $\frac{T_s}{T_{\mathrm{na}}}$ as a function of $\frac{P}{P_r}$, namely
$$
\frac{T_s}{T_{\mathrm{na}}}
= \exp\left(0.174205\cdot(\tfrac{P}{P_r})^{0.150263}
+ 1.83121\times10^{-5}\cdot(\tfrac{P}{P_r})^{1.04193}\right)
$$
gives a coefficient of determination $R^2 = 0.9999$. Wow!
Is this surprising?
Um, no. What they've discovered there is most of all the ideal gas law. That connects pressure and temperature in a pretty rigid way.
So far, everything is... ok. Nothing actually new (just long-known physics, re-discovered in an inefficient way). The trouble only really starts with the conclusions.
Nikolov/Zeller argue that this excellent fit, and the much worse fits of similar models relating other quantities – in particular, greenhouse-gas concentrations – mean that it's actually only the pressure which influences the surface temperature, and that GHG are irrelevant.
But that's completely and utterly fallacious!
Again: it's not surprising that the ideal gas law holds – rather, it would be extremely surprising if it didn't. So, the $T_s(P/P_r)$ fit tells up about as much as a fit of fall-times of weights from different heights in vacuum: nothing, they just reaffirm the long-known Newton laws of motion.
And it does not mean that “temperature is only influenced by pressure”. On the contrary: just as much, pressure is influenced by temperature – if you somehow lowered the mean temperature of a given planet, surface pressure would drop as well‡.
Both pressure and temperature also depend on other quantities though. Notably, yes, on greenhouse-gas concentration. But unlike the gas law, the mechanism is a bit more complex and can't be captured properly¶ by fitting such a primitive model to a few measured points. Instead, it requires 3D climatology models, which incorporate all of these quantities, and not just in a global-mean sense but in a detailed, latitude- and altitide-dependent way.
I said I'd come back to the scenario of an atmosphere of significant density but without greenhouse gases. Such an atmosphere would keep spatial and temporal variations limited, but wouldn't influence the radiation processes, i.e. this is the case where the surface temperature would actually be similar to $T_\mathrm{e}$. How is this still compatible with the Nikolov/Zeller model 12? Well, again that model is mostly just the gas equation. Such an atmosphere would still obey it. The higher reaches would be colder than the surface due to adiabatic expansion. Really, this scenario isn't very exotic in terms of fundamental physics, but it illustrates that GHG are absolutely essential for a climate like Earths. The Volokin/ReLlez paper just proves that $T_\mathrm{e}$ isn't relevant for bodies without any atmosphere at all, but that's hardly interesting for any discussions regarding Earth's climate.
†The paper specifically discusses the Hölder inequality, which is somewhat curious because that inequality actually states that there is, for power functions, a relation between mean-of-the-function $\langle f(x)\rangle$ and function-of-the-mean $f(\langle x\rangle)$, namely, the latter gives an upper bound to the former. The Hölder inequality does not actually imply that $\langle f(x)\rangle$ lies beneath that upper bound – it could just as well instead be smack on the bound; but it's true that there are quantities for which it is substantially below.
‡At that point, I am oversimplifying. There's a whole bunch of different scenarious what could happen when you alter a planet's surface temperature (even assuming you managed to do this in a way other than directly tweaking atmosphere properties). The pressure-drop case is what happens when it gets so cold that the formerly atmospheric gases condense onto the surface; this is relevant to e.g. Mars and Titan. In other planetary bodies, other mechanisms would step in; in either case however, you could be sure that the ideal gas law continues to hold.
¶Ironically enough, Nikolov/Zeller's Model 7 actually hints that greenhouse concentration does correlate to temperature, even though it lacks the quantities that would be needed to make the fit tight.