Does sulfoxide H₂SO exist?
This is not an answer to the question "can the sulfoxide be synthesized in significant yield" (and under which conditions). Instead, this is a "naive" prediction based on the data you provide and two assumptions: 


The predicted thermodynamic values in the articles you cite are accurate at $\pu{298 K}.$
The heat and entropy of formation of (2) and (3) from (1) are constant over $T.$  


From the references we can build a data table of energies and entropies relative to "the ground state" $\ce{HSOH}$ assuming the $\Delta G^\circ$ values correspond to $T = \pu{298 K}:$

$$\begin{array}{lccc}
\hline
\text{Compound} &amp; \Delta G/\pu{kcal mol-1} &amp; \Delta H/\pu{kcal mol-1}^* &amp; \Delta S/\pu{cal mol-1 K-1} &amp; K_\mathrm{eq}\\
\hline
\ce{H2SO} &amp; 25 &amp; 15.8 &amp; -31 &amp; 10^{-19} \\
\ce{H2OS} &amp; 34 &amp; 38.2 &amp; 14 &amp; 10^{-25} \\
\hline
\end{array}$$

I include the order of magnitude of the equilibrium constants $K_\mathrm{eq}$ only to illustrate their ludicrously small values (and these are upper estimates!).

Could you find a temperature that would make $\Delta G^\circ&lt;0$ for the reaction to $\ce{H2SO}$? The answer is no.  To see why we turn to the enthalpies. Since the reaction is endothermic you might hope that an increase in $T$ would help, but the accompanying entropy change is negative, so there is no way you can sufficiently increase the temperature to drive the reaction forward. The highest equilibrium constant you could hope for is $K_\mathrm{eq} = \pu{1.7E-7}$. 

Things are not so bleak for $\ce{H2OS}$: the reaction is also endothermic but the entropy change is positive. At about $\pu{2800 K}$ $K_\mathrm{eq} = 1,$ and if we increase $T$ sufficiently, we can crank up $K_\mathrm{eq}$ to any arbitrarily large value. 

This is of course not remotely exhaustive (therefore "naive") but hopefully a start to answering your question. 



* Quoting your second reference: 


  At the CCSD(T)/CBS limit and including corrections for scalar relativistic, spin orbit and core-valence correlation effects, the estimated enthalpies of formation are $\pu{−28.1 ± 1},$ $\pu{−12.3 ± 1},$ and $\pu{10.1 ± 1 kcal/mol}$ for $\ce{HSOH},$ $\ce{H2SO}$ and $\ce{H2OS},$ respectively.