I can contribute some details here as well.
Moon of a Larger Gas Giant
While gas giants do have huge magnetic fields that deflect a lot of radiation, they also create tons. Jupiter's moon, Io for example is in a radiation belt strong enough to give a human a lethal dose in a very short timespan. I can elaborate on xpian's ideas about heat generation as well. Jupiter generates heat because it's still contracting from its formation. That gravitational energy is radiated out slowly. However, this energy does little to warm anything except Jupiter itself.
Additionally, unless your moon is a captured asteroid, it's almost guaranteed to be composed primarily of ice. Most of the gas giant moons are. This has advantages, though. The tidal forces that stretch Io are also suspected to be stretching Europa and Callisto, possibly creating a liquid ocean beneath a surface of ice. Sooo, if you want an aquatic race, gas giant moons are excellent. But xpian is right. You could easily have something like an earth-sized planet an acceptable distance from a gas giant to support life. With a thick enough atmosphere, you could even keep it relatively warm.
Twin Earths
Entirely possible. In fact, Earth and its moon are considered to be a "double planet" because they're so close in size (astronomically speaking). This likely would cause tidal locking, but wouldn't necessarily mean some parts of each planet get no sunlight. The moon is tidally locked to earth, but because its axis is tilted, it doesn't create an eclipse every time it orbits.
Axial Tilting Periodical Movements
Yes. Theoretically possible. What you're describing is an effect called "precession". Earth does it at a small angle on a 50,000 year cycle. Think of it like a spinning top that starts to wobble. We don't see this mess with the seasons on Earth because that 50,000 year period is far slower than the 365 day period for a year. But IF you had a planet that was precessing much faster than it was orbiting, you could get some wonky effects. However, any planet precessing that fast would sort of even itself out. So the only way you could do this would be to bend reality and have the planet precess very quickly but have a very far orbit. Science fiction, right!
I can add a few other crazy ideas, too.
Interesting Exosolar Planets
Some of the planets we're theorizing have some crazy properties. Planets tidally locked with their parent stars so that one side is permanently dark, the other bright. This creates turbulent wind patterns at the terminator (the border between light and dark). You could then have a planet with permanent areas of twilight with permanent violent storms.
We've also theorized that there are "carbon planets" composed almost entirely of carbon. These planets could literally have volcanoes that erupt diamonds.
Rogue Planets
It's hardly impossible to have planets that get flung away from their parent stars and sent into deep space. These rogue planets would literally just wander between stars, almost permanently dark and frozen. But you could have life living deep under the surface, surviving on heat radiating from the core. I think there was a Star Trek Voyager episode that did something with a rogue planet.
If your axial tilt is stuck vertically (which, if I understand what you're getting at would mean NO tilt), then you pretty much won't have seasons. You'll get a planet with weather the same all year round. Farther north you go, the colder it gets, but that'd be about it. If your trajectory is elliptical enough, you will get seasons, but they'll affect the entire planet at once, rather than the north-south hemisphere opposites we get on earth.
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