Under what conditions may a pop III star undergo a supernova, i.e., only certain masses, composition, etc?
The exact mass range, composition, rotation etc. necessary to for a star to go supernova or to undergo a direct collapse are still uncertain (see for example O'Connor & Dott 2011, Fryer et al. 2012, Ertl et al 2016). If this is uncertain for PopI and PopII stars, you can imagine that for PopIII, lacking any observational evidence, the problems are even worse.
But, even if the exact conditions are not known, the processes that may lead a PopIII star to explode are relatively well understood and were already described in Heger et al 2003 that you quote.
Why PopIII stars are different from solar metallicity stars
Having a lower metallicity means less efficient stellar winds (Grafner & Hamman 2008, Rickard et al. 2022), so a less pronounced mass loss. This means that the helium core of PopIII stars might not be eroded by winds, even for very massive stars ($>100 M_\odot$). Thus, PopIII stars might build massive helium cores, which is not possible for their high metallicity counterparts. And the mass of the helium core is a key factor in determining the final fate of the star.
Core collapse supernovae, direct black hole formation of pair instability
Quoting the numbers from Table 2 of Heger et al 2003, a star that at the onset of collapse has an helium core of at most $8 M_\odot$ will explode as a core collapse supernova. If $8 M_\odot < m_{He} < 15 M_\odot$ the supernova still occurs, but only little is ejected and most of the material falls back into the black hole. Helium cores in the range $15 M_\odot < m_{He} < 42 M_\odot$ lead to a direct collapse into a black hole, with no supernova explosion.
Quoting the mass ranges from Section 4.4, above $42 M_\odot$ things start to change. The helium core is now massive enough that it can reach a temperature of about $10^9 K$, at which the production of electron-positron pairs becomes efficient. This soften the equation of state to an adiabatic index below $4/3$, reducing the pressure support and inducing the contraction of the core. This process is called pair instability and is thought to occur only for very low metallicity stars that are able to form such massive helium cores. If the helium core has a mass $42 M_\odot < m_{He} < 65 M_\odot$, the energy produced during the contraction is able to reverse the collapse. The star will then undergo a series of violent pulsations (Yoshida et al. 2016, Woosley 2017) until it has lost enough mass to stabilize the pair production, and will (likely?) eventually collapse into a black hole without further explosions.
If instead $65 M_\odot < m_{He} < 133 M_\odot$, the first pulsation is strong enough to completely unbind the star, that explodes in a Pair Instability SuperNova (PISN), leaving no remnant behind. These highly energetic events are thought to be important to the early chemical evolution of the universe (due to the large amount of metals expelled) (de Bennassuti et al. 2017), but have not been conclusively observed yet, although there is some hope in the next generation of telescopes (see for ex. Moriya et al. 2022, Tanikawa et al. 2022). Since stars in this mass range do not produce a black hole, it is expected that we should observe a gap in the black hole mass distribution, called pair-instability gap, but there is currently no evidence for this (The LVK collaboration, 2021).
Above an helium core mass of $133 M_\odot$, the star is expected to collapse again into a black hole, without any explosion.
A nice way to visualize all these mass ranges is this figure from Spera & Mapelli (2017) that shows the relation between the initial mass of a star (ZAMS stands for zero-age main sequence) and the mass of the black hole remnant produced. Each curve corresponds to a different metallicity, from $Z = 2 \times 10^{-4}$ to $Z = 2 \times 10^{-2}$. Note that the x axis is the initial mass of the star, not the final mass of the helium core. This means that this figure is assuming a relation between initial mass and final mass of the helium core, which is affected by additional (large) uncertainties.
Look, for reference, at the purple dashed line ($Z=2 \times 10^{-4}$) which is the closest to PopIII metallicity. For $m_{ZAMS} < 35 M_\odot$, the core collapse supernova occurs, most of the material is ejected, and a relatively low mass black hole is formed. In the range $35 M_\odot < m_{ZAMS} < 60 M_\odot$, there is a direct collapse into a black hole, in fact $m_{ZAMS} \approx m_{rem}$. For $60 M_\odot < m_{ZAMS} < 110 M_\odot$, pulsational pair instability considerably reduces the final black hole mass, until, for stars with $110 M_\odot < m_{ZAMS} < 230 M_\odot$, a PISN occurs and the mass of the remnant is zero. For initial mass greater than $230 M_\odot$, the direct collapse resumes.
Keep in mind that these numbers are model dependent and considerably more uncertain than the figures for the helium core mass. Also, all the above is not taking into account rotation, which may be of great importance (Meynet et al. 2009, Boyouan et al. 2020)
Second questions
Does (the core of) a pop III star that undergoes a supernova form a proto-neutron star before forming a black hole?
Given the above, I would say no, the mechanisms for explosion is not electron-capture, but pair production.
Third question
The paper you quote is talking about supermassive Pop III stars, an hypothetical kind of star with thousands of solar masses that could be formed as the result of gravitational instabilities in primordial clouds that are not able to fragment through $H_2$ cooling, because a near source of Lyman-Werner radiation is dissociating all the molecular hydrogen. The result is thought to be a monolithic collapse of the cloud that leads to the direct formation of a supermassive black hole in the range $m \approx 10^{4-5} M_\odot$. The authors of the paper argue that for a narrow mass range, instead of forming a black hole, the cloud could form a supermassive star, and that such star would explode in an extremely energetic event. The reasoning behind the apparent self-contradiction in the statement you quote is, in my opinion, the following:
if we detected an energetic event such as the one we are predicting, this would be a confirmation that supermassive stars can form from a direct collapse of a primordial cloud. According to the same mechanisms, similar clouds with a little more or little less mass would instead directly form a supermassive black hole. Thus, the observation of such a supernova would be an indirect confirmation that supermassive black hole heavy seeds can indeed form through direct collapse of the primordial cloud. This would help in addressing the open problem of how the observed supermassive black holes at high redshift could form in such a short time.
