I'm going to assume that the planet may be both Jupiter-like, and Saturn-like to give us more real world examples to work with. I'm also going to assume that said gas giant is at 1 AU from the sun so there is sufficient heat for Earth-like life. Imagining life in a cold Europan ocean is too broad for this topic. As for gas giant formation that close to the sun, there is ample evidence for that in exoplants. I don't think that we understand solar system evolution well enough to say that Jupiter in Earth's orbit is unrealistic.
Possibility of formation
Jupiter's moons are assumed to be the four left of a larger set of moons. As Jupiter's ancient moons formed, the proto-planetary disk around the sun was still relatively dense. This cause the moons to lose velocity and spiral into Jupiter. The four remaining were far enough away that they maintained orbit until the sun sucked up or drove off most of the free dust. So if (big if) your Earth-sized moons developed far enough off, then they could have formed. I suspect a larger planet than Jupiter would be needed to form moons of that size though...I don't know.
Saturn's inner moons apparently underwent a cataclysmic collision around the time of the dinosaurs that destroyed them. The debris then reformed into Saturns larger inner moons (basically all of them except Titan and Iapetus). You don't want that to be happening on your moons...unless you want four to destroy and reform while life is evolving on the outermost moon...the denizens of which could colonize the inner moons. There is no explanation as to why these inner moons collided...so there's that.
So the answer to, "could these moons form" is...maybe? Lots of mystery here. But the larger the gas giant the more likely it is to happen, in my opinion.
Hill Spheres and Moon's Moons
The Hill sphere is the region of space where a planet or moon can keep a satellite. Earth's is 1.5 million km, while the moon is 384,000 km away. So close the Jupiter, the hill sphere is smaller. An Earth-like planet in orbit of Jupiter at the orbital distance and eccentricity of the four large moons (Io, Europa, Ganymede, Callisto) would have a Hill sphere of 42, 36, 87, and 152 ,000 km. This is significantly smaller than the distance to our own moon. However it is larger than the distance to Deimos (23,000 km) the farther of Mars' two moons. So you could have Moon's moons, but they'd be smaller than our moon, and closer in too.
The farthest out large moon of Jupiter or Saturn is Iapetus, at 3.6 million km. If Earth was at this distance from Jupiter, its Hill radius would be 352,000 km. If it were circling Saturn (a smaller gas giant), it would be 526,000 km. This is the only configuration that could conceivably support a moon at Earth distance.
This just covers Hill spheres, there are plenty of other orbital dynamics that could disturb secondary moons forming around primary moons. In general, let me express my doubt that any of the moons could have their own moon, and let me be especially doubtful that any of those moons could be larger than a tiny moonlet.
The co-orbital moon idea is also interesting. Tethys and Dione (two moons of Saturn) have 'Trojan' moons in leading and trailing orbits. These moonlets are from 2–35 km in size.
Radiation
Jupiter's satellites get...a lot. Io gets 3600 rem/day, Europa 500 rem/day, Ganymede 8 rem/day, Callisto 0.01 rem/day. 1000 rem is deadly, 100 rem will give you radiation sickness, 0.1 rem is on the order of a normal dose on Earth over a YEAR. So on Callisto you'd be fine, on Ganymede you'd die in a month, Europa a couple of days, and Io immediately.
Saturn's satellites on the other hand don't get very much. Jupiter has a powerful magnetic field, but that causes the radiation. Your gas giant will need a weak field like Saturn. On the other hand, the planets' inhabitants need to be defended from cosmic radiation sources, so each planet will need its own magnetic field. Alternatively, one the planets of which life developed could have magnetic fields, while the rest are colonized and have structures built underground to protect them from cosmic radiation.
Looking at moons from other moons
I calculated all these myself with a spreadsheet and trigonometry, so there may be some errors. Here are some reference numbers: The full moon from Earth is 31 arcmin, which is just over half a degree in the sky. The sun is about the same, although it is obviously much brighter. For the large satellites of Jupiter and Saturn, the largest a planet appears is Jupiter from Io at 549 arcmin or almost 9 degrees of the night sky. Another way to put this is 1/20th of the way from horizon to horizon or 18 times the size of the moon. Thats almost too big to comprehend. The smallest a planet appears is Saturn from Iapetus, at 56 arcmin, or still twice the size of the moon. All your moons will have significant light from the gas giant.
If Earth-sized planets were in the orbit of Io and Europa, at closest approach they would appear to be 175 arcmin from each other, or 3 degrees, or 6 times the size of the moon. If Earth sized planets had an average orbit around where Iapetus is, they would appear to be 6 arcmin from each other, 1/5 the size of the moon.
Colonies on small moons
One word: don't. Humans live at 1 g of gravity. Gas giants don't have surfaces, Venus has .9 g at the surface, Mars and Mercury have 0.38g. Our anatomy and physiology is designed for 1 g, it helps our bones calcify properly, helps our blood circulate, and does countless other things we may not even have discovered yet. Humans maybe could live a lifetime on Mars and Mercury. It is very unlikely they could live a lifetime at an even lower gravity. Luna, Titan, and the 4 Jovian moons have gravity of 0.12–0.18 g. Pluto, Triton, and various Kuiper belt objects are below 0.10.
A medium sized moon (like Rhea of Saturn, say) might have 0.027 g. This is way not enough for a human to live, but annoyingly much if you want to, say, launch things into space. Humans are better off living in space colonies that generate artificial gravity by rotating. If you want to exploit a moon, you are much better off with one that has a much lower surface gravity. Phobos, for example, has a radius of about 11 km, and a surface gravity of .0006 g, but still has $10^{16}$ kg of usable materials to mine.
