I have been doing some late-night wikipedia binges on this matter, so I consider myself qualified to answer.
> Is there a comprehensive list of all the planets, dwarf planets, moons, etc. that we suspect may have liquid water?
Most large moons orbiting the gas giants are suspected to have subsurface water oceans, most prominent among them Europa. Ganymede, Enceladus, and Triton are other honorable mentions: https://en.wikipedia.org/wiki/Planetary_oceanography#Natural...
> How many of them have liquid water without being super saturated with salts? What is the level of confidence we have?
We don't know the composition yet, just that a salty ocean supports the evidence we've seen. For example, Europa interacts with Jupiter's magnetic field in such a way that makes sense only if there is a large amount of electrically conducive material inside Europa. Given what we know about its formation, metallic stuff is unlikely. A salty subsurface ocean is a more suitable explanation. We'll know once Europa Clipper gets there (one of its missions is to 'taste' the ocean by flying past really close).
There are other reasons to suspect a subsurface ocean, such as signs of a geologically active body (like Enceladus), which in turn implies there is enough heat inside to have an ocean.
> Which are most likely to bear life given our current hypotheses around the origins of life?
Europa, maybe Enceladus. Note that our search for life is heavily guided (biased?) by our understanding of how we think life started here on Earth.
Specifically, we needed these ingredients (non-exhaustive):
* a liquid medium to act as shelter and transport medium (liquid water ocean)
* a source of energy, something that life can 'eat' (hydrothermal vents in the beginning, sunlight later)
* protection from space-borne threats (atmosphere, magnetic field)
* a home that isn't geologically dead (like the Moon, or Mercury), but also isn't too active (like Io)
Europa checks most of the requirements:
* suspected subsurface ocean
* geologically active. This is important because Europa and other moons are too far away for sunlight to be a big source of energy. There has to be another source. Radiogenic heating (released by radioactive isotopes) and primordial heating (left over from its formation) aren't enough for moons to sustain liquid water (especially 4.5 B years after formation). This means tidal flexing is what we should be looking for -- something we know Europa has.
* Europa's thick icy crust blocks pretty much all radiation. Barring a thick atmosphere (which only Titan has), a subsurface ocean underneath an icy crust is the best life can hope for out there, for protection against radiation.
* We don't know much about what goes on underneath the surface. If the bottom of the ocean contains geothermal vents, then that could harbor earth-like life.
Saturn's moon Enceladus checks many of these same boxes (tidal heating, abundant water, signs of a young surface indicative of geological activity).
JUICE and Europa Clipper missions should fill in many of these gaps, and help confirm our understanding.