your question maybe would take better place at the astronomy-section of stackexchange... or maybe not.
Anyway, while math is my enemy too, I came across such questions while programming a tool for astronomy purposes.
First of... design a stable binary system, before you place your living beeings. There are many out there in space in reality, so its plausible to assume these can hold planets with lifeforms too.
But one point creates headache at my side: you want your moons planet orbit both suns. He would orbit the... uh, what was this name... barycenter (?) of these suns in that case, getting pushed around by one of the two suns on a regular base... hell, I don't want to this planet, when it and his two central bodys form a conjunction. Here the math kicks in, and I have to fall back to speculate, that it might rip apart a big gas giant if he needs to be close enough to be inside the habitable zone of this system.
But! Most binary and trinary systems don't have their suns do a close dance around each other. Prove me wrong, but I think in most cases the second sun acts as a remote planet, that happens to do stellar nucleosynthesis. So the second sun orbits the central sun faaar away - think about the oorthian cloud, or what is name was - and have to use what was left of material when the first sun collected itself. So it might be a red dwarf.
Another thing - two real close dancer suns will eat each others material - the bigger one can steal stuff from the smaller one, if close enough. That might result in a pretty spontaneous combustion of the small one. But to make this work, the big one need to be pretty big (H or O class? Hmmmm... no, any class super or hyper giant may be sufficient). At least the "bright" ones tend to have pretty short lives... short enough, to go supernovae and destroy their small follow star before any life could develop.
Well, I would recommend a decent stable system, where star number two is a far orbiter. Feel free to update my knowledge about this if I recalled this wrong.
Big gas giant, even bigger than Jupiter? Don't go to far... let me kick my brain... but I think over 1*10^28kg of mass this thing will fuse some low level nuclearsynthesis. Other words: you might get a brown dwarf. They consume a special element I can't remember (something containing a y?), don't do what real stars do but would be much warmer than a common gas giant.
Even a Jupiter size gas giant will be a bad place to orbit around. Ever heard of jupiters music? Feel free to google this :) But anyway, jupiter size gas giants may be pretty ray bursters. This may encourage evolution, but wouldn't be a pleasureful place to life for human like creatures. Okay, I may overestimate the power of the emitted rays, but that something you might think about.
A planet this big will catch most stuff in its orbit. I think astronomers call this "clean its orbit". Due to many millenias every piece that came across the gas giant will be slung away. It can go away, it can become a new satellite, or it may impact your moon. When life there did grow naturally, most of these planet killing stones should have gone... but maybe not all.
Oh, same count for your moon. It will clean its path, so if its an earth-like stone planet, you probably won't have any rings at this gas giant.
Ehm... well... you know... to get a orbital period of one earth year for a planet (moon) about the same size (and mass!) as our beloved earth, you need a earth-like orbit.
So... about 150.000.000 Kilometers. If you have a sun-like star AND your gas giant inside the habitable zone, this would get... warm in summer. Very warm. Well, you moon-earth would slam inside the central sun or get catched by it in a mercury like orbit.
Hm... one point I'm not sure about right now is... how much closer to its central body a satellite may come when taking the central bodys mass into account. I think you won't need the whole 150.000.000km, but... hm... 15 Million maybe. Argh, I have no access to my program here, but I think someone who can calculate this will pop up in this thread pretty soon :) Or use one of the free calculators in the internet... but to make a bet: to have an orbital period of one year for a planet of a earth like mass in 15 million kilometers semimajor axis, you would need a central body of 1/100 sun masses... that is... a brown dwarf? sigh there is a reason I need a program to guess stuff like this. whats 1.9 x 10^30 / 100 ? 1.9 x 10^28? well, that would be pretty brown dwarf, wouldn't it?
But! but, but but... You want a bound rotation of your moon. That means your moon needs to be close to you gas giant. close enough, that the gravity of the gas giant can friction break your moons rotation using waves, air and even continental plates. So... this close, you will receive lovely amounts of hard em rays and get a forced orbital period of... some weeks in best.
But you say "lol its magnet field would neutralize this noob"? But we need to take care of the seismic activity and vulcanism, which requires a moon with death core.
Correction: a earth size planet is likely to have a liquid core very long. Even Mars and Venus should have one, Mercury... don't know. Hell, didn't the biggest of Jupiters moons claim a magnet field?
It looks like all your creatures can do is embrace the natural disasters. But anyway, once rotation got bound, things will calm don I think.
How to neutralize the temperature? Okay, I'm off here. If you don't want to get your moon scorched by the central sun, it should be at the outer range of the habitable zone. But... here a brown dwarf would be useful, because it does emit heat I think. So the side bound to the gas giant / brown dwarf may receive enough energy to allow liquid water, while the other side... you may install a ice princess over there.
To expect what the seasons be like, I would like to wait for what moon - planet - binary sun constellation you vote at the end. Because it will matter a lot.
To answer you last question... maybe. If you can accumulate enough energy from what sources available to get about the same level as we do have on earth, it may work. Get this energy from a sun, or two, a brown dwarf, an active gas giant... but don't get too much, or you get a Venus, and too less will create you a Mars.
Then avoid heavy hard rays, so no blue or white central star and a peaceful gas giant. To avoid heavy changes of temperature and climate, you need to stick in a narrow piece of your habitable zone, so no wide orbits around that gas giant (needed for bound rotation too), which means you have to forfeit the six month day/night I'm afraid of.
But you can give your gas giant a slight elliptical orbit, so it will bring its moon closer to the sun half of its year, which may be... well, I didn't talk about this, but if your gas giant wants to be stable, you may need orbit him in a distance that take many years for a single circuit. That would make a maya-civilization happy, because their calendar would work pretty well telling when the next "summer" will arrive.
So. I'm out of words for now. And interested in your decision about this system at the end.
Have a nice day
Edit: Oh well, that answer in that first comment to your questions is 10^28 times better than my stuff... :( keep with that