One of the most habitable moons of Jupiter is Europa:
A base there would also offer up a spectacular view: Jupiter would
look almost 24 times larger in the sky than Earth's moon appears from
the planet's surface. "It wouldn't be a boring place to drink your
afternoon tea," Vance said.
However, residents of Europa would have to deal with the blistering
cold. The temperature averages about minus 260 degrees Fahrenheit
(minus 160 degrees Celsius) near the equator and minus 370 F (minus
220 C) near the poles, Pappalardo said.
Aside from the radiation and cold, a few other dangers exist on
Europa. Modest ice quakes — which might feel somewhat like an
earthquake on Earth — could occur every now and then, and you'd
definitely want to watch out for giant plumes of water that could
violently shoot out of the ice. There would also be a small chance
you'd be hit by a small meteor or particles from space.
Europa's gravity is only about 13 percent of Earth's, resulting in a
virtually nonexistent atmosphere. Because of this, Europa, like
Earth's moon, has no weather, wind or sky colors. And walking on
Europa would feel similar to walking on Earth's moon.
If you went for a stroll, you'd probably want to visit Europa's "chaos
terrain," where the normally flat ice has been fragmented into jumbled
blocks. "It might look like you're in Monument Valley," Pappalardo
said. You'd also want to see the moon's depressions and ice domes,
which are up to a little more than a half-mile (1 kilometer) high.
But, neither Europa nor Jupiter are as big as a moon around a gas giant or a gas giant can get. The maximum possible mass of a gas giant before it ignites into a brown dwarf star is 11-16 times the mass of Jupiter and the diameter if Jupiter is as big as a gas giant can get (adding mass would only increase it density).
But, a moon around a large gas giant can get as large as the planet Neptune ("surface" gravity 110% of Earth's) or Uranus ("surface" gravity of 88% of Earth's). And, while Jupiter has 60 moons, nothing in the laws of physics (ideally helped along with a little orbital engineering) prevents it from having fewer, larger moons than it does. It would be tricky to figure out how to keep any of the planets from colliding into each other for a very long period of time, but not impossible by any means.
It would be more realistic, however, to have six more or less habitable moons and at least a dozen or so more small moons that were not habitable.
With larger moons, the gravity and atmosphere problems can be solved. The right rotational speed on its axis and right tilt of its axis and atmosphere combined would moderate the temperatures greatly. You would need the gas giant to be much closer to the Sun, or it would have to actually be a brown dwarf rather than a gas giant, however, to make the moons sufficiently warm.
Their speed of rotation around Jupiter respectively would have to be fast enough to prevent the moons from getting too cold when they were in Jupiter's shadow relative to the sun (although terraforming with a particularly thick atmosphere could also help with this issue), if you want to avoid the very strange and possibly unstable dynamics of having the entire system on a huge axial tilt relative to the star around which it orbits.
The ring of rocks and minerals wouldn't be very habitable, but there would be no problem with having one, and indeed, you would expect one.