1) How far out into the galaxy do our own signals travel before they dissipate or become no longer recognizable as being "not natural?" How far away are we visible to anyone who is looking? And when are we visible there?
That first one is difficult to answer. It's a matter of resolving a signal awash in noise created from natural planetary, astrophysical, and galactic processes. And extraterrestrials might have come up with some pretty clever ways to do such a thing, but we've still gotta consider the time delay of the signal. Humans have only been making radio waves for ~150 years, so only extraterrestrials within ~75 light years away would have had time to receive them, respond, and send a signal back at us (if they chose to, of course). Anything further away than that, well, we wouldn't know if there's something there yet. And it appears that there are plenty of planetary systems (likely with moons, too!) within 75 light years, but I'll bet that intelligent life is quite a bit more sparse than that.
Now, if you're talking optical detection, and presume that extraterrestrials are looking for man-made structures, I think the pyramids might be our best bet. So that's ~4500 light year radius of signal, only 1/10th of the radius of the Milky Way. And we'd have only just heard back from anything within ~2250 light years. To give you an idea of where we are with optical wavelength resolution, I'd estimate that it could take us another 200 years-ish before we're able to resolve pyramid-size surface features on the planets orbiting Proxima Centauri, and that's only ~5 light years away. And hey, if they'd seen "only" dinosaurs (my quotes/italics, for emphasis), that would have absolutely warranted sending shittons of probes here. Probably a continuous stream of probes. Definitely an argument against aliens, at least in the Milky Way, and intergalactic exploration seems downright impossible. Sucks :(.
2) Then there's the question of how would you build and launch a spaceship from a world where you live in liquid under a mile of ice?
Shoot, they could figure something out, I have faith in our hypothetical intelligent species!
3) Would the moons of such a gas giant, if close enough to the planet, benefit enough from the magnetic field to be used instead of building a space station?
Maybe! I just checked, and although Jupiter's four largest moons (Io, Ganymede, Europa, Callisto) all intersect the deadly radiation belts, there are some captured asteroids that supposedly orbit in the sweet spot just above the cloud tops. One of those with a small orbital inclination (so that it doesn't intersect the high latitude portion of the rad. belts) might be perfect. And I'd wager that systems like Jupiter and it's moons are relatively common. We already know that Jupiter-like planets abound, but don't know much about how commons moons are, and the distribution of magnetic field strengths, orientations, stabilities, etc. Of course, the planet has to orbit within a habitable zone... ugh, so many variables.