In this answer, I’ll attempt to address two primary concerns impacting the habitability of your moon:
- atmosphere retention
- absorption of solar radiation
You will undoubtedly have to tweak the parameters of your planet in order to get the desired weather patterns. However, these two factors seem most important with respect to habitability.
Before getting into the weeds, here’s a list of definitions for variables I’ll use:
- $R_m$, the moon’s radius
- $M_m$, the moon’s mass
- $L_{s}$, the combined average luminosity of the three-sun system
- $D$, the distance of the planet-moon system from the three-sun system
- $G\approx 6.7\cdot 10^{-11} \space\text{Nm}^2/\text{kg}^2$, the gravitational constant
- $k\approx 1.4\cdot 10^{-23}\space \text{J}/\text{K}$, the Boltzmann constant
Alright, let’s go! (Note: I’m bound to have made some computational error somewhere below. Hopefully it doesn’t affect my estimates too much, and they’re still within the right order of magnitude. Bonus points if you find a mistake!)
Atmosphere retention
No matter how massive or cold your planet is, it will always continuously lose some of its atmosphere (as long as this atmosphere is gaseous). This is because not all of the atmospheric gas molecules have the same speed - their speeds are random, following the Maxwell-Boltzmann Distribution. At all times, some of the molecules will be moving fast enough to escape. The question is - how long do you want your atmosphere to last?
The escape velocity for your moon is approximately equal to
$$v_{\text{esc}} = \sqrt{\frac{2GM_m}{R_m}}$$
and the root-mean-square velocity of gas molecules in a gas of temperature $T$ is equal to
$$v_{\text{rms}} = \sqrt{\frac{3kT}{2m}}$$
where $m$ is the mass of the gas molecule in question. You certainly don’t want $v_{\text{rms}}>v_{\text{esc}}$, or your whole atmosphere will be gone in an instant. So, at the very least, you need
$$\sqrt{\frac{3kT}{2m}} \lt \sqrt{\frac{2GM_m}{R_m}}$$
or, for a molecule of diatomic oxygen,
$$\frac{M_m}{R_m T} \approx 2.92\cdot 10^{12}\frac{\text{kg}}{\text{m}\cdot\text{K}}$$
For a moon the size of Deimos (which is almost certainly much smaller than yours) and with an average surface temperature equal to Earth’s, the LHS of this inequality is approximately $8.3\cdot 10^{8}$. That’s well below this rudimentary upper limit - so far, so good.
Let’s get a little more nitpicky. Remember what I said before about how some of your planet’s atmosphere will always be escaping?
Assuming the atmosphere’s depth is negligibly small compared to the planet’s radius, we have that the surface area of atmosphere exposed to space is approximately $4\pi R_m^2$. According to the Maxwell-Boltzmann distribution, if $T$ is the average temperature, then the proportion that have achieved escape velocity at any given time is equal to
$$\begin{align}\alpha_{\text{esc}} &= 2\sqrt{2\pi}\int_{\sqrt{GM_m m/kTR_m}}^\infty v^2 e^{-v^2}dv\\ &= \frac{2\xi e^{-\xi^2}+\sqrt{\pi}\text{erfc}(\xi)}{4}\\
&\sim \frac{\xi e^{-\xi^2}}{2}
\end{align}$$
for reasonably small values of $\xi$, where $$\xi=\sqrt{\frac{GM_m m}{kT}}$$
As an estimate, let’s use the Moon’s mass and radius and Earth’s surface temperature (and consider diatomic oxygen molecules). This yields approximate values of
$$\xi\approx 18.5$$
$$\alpha\approx 2.13\cdot 10^{-148}$$
Yowza, that’s a tiny value of $\alpha$! The volume of atmosphere that would escape over the course of $t$ seconds would be approximately equal to
$$4\pi\alpha R_m^2 v_{\text{esc}} t$$
But I’m not going to proceed further with the calculations. The value of $\alpha$ is so microscopically tiny that it will basically overwhelm the other factors in the above expression. Looks like your planet’s atmosphere is probably safe!
If you really want to make sure your atmosphere is secure, I’d recommend the following additional precautions:
- Make your planet nice and dense. This keeps $R_m$ low while driving up the value of $M_m$, which will make $\alpha$ even tinier.
- Give your moon and the planet it orbits a hefty magnetic field to deflect atmosphere-destroying cosmic rays.
Absorption of solar radiation
Now for the easy part! This won’t be nearly involved as the above.
I claim that any given point on your moon’s surface spends about $1/4$ of the time in the daylight and $3/4$ of the time in the dark, under the following assumptions:
- no tidal locking, as stated in the question
- the moon’s orbit is independent of position of the planet it orbits around the sun
- the gas giant is massive compared to the moon
- the three stars in this ternary star system are relatively close to each other and very far away from the planet and its moon
Why? Well, about $1/2$ of the time, the moon is on the opposite side of the planet, so it receives no light. When it is on the lit side of the planet, only $1/2$ of the moon’s surface is lit at any given time. Thus, for any point on the moon’s surface (poles excepted), it is lit about $(1/2)(1/2)=1/4$ of the time.
This means that, in order to maintain an Earth-like climate and temperature, something must compensate for this increased duration of night-time. Here are some suggestions:
- Greater amount of solar radiation. There are three stars in the system, after all.
- Increased luminosity $L_s$ of the stars.
- Smaller distance $D$ from the three stars. It wouldn’t have to be much smaller, though, since intensity at a distance $D$ is proportional to $1/D^2$.
- Lower albedo, to avoid reflecting away solar energy.
- Lots of greenhouse gases to help trap solar radiation energy.
Here are some other non-sequitur speculations about what your moon might be like:
- You mentioned that you didn’t want there to be any tidal locking, but if there’s any significant amount of liquid water on the planet’s surface, the gravitational pull of the gas giant will exert significant force upon it. At the very least, this could cause some very extreme tidal rising and falling (exacerbated by the planet’s low gravity), creating vast tidal zones on the planet’s surface.
- As mentioned above, the day-night cycle on the moon will be wacky, nothing like the regular half-day-half-night cycle of Earth. There will be a long stretch of darkness (when the moon is behind the planet), followed by a series of day-night cycles whose length depends upon the rotational velocity of the moon, and then a return to darkness. I wonder how this will affect the circadian rhythms of animals and photoperiodism of plants on the surface?
- Since the moon spends a significant amount of time on the dark side of the planet, freezing/thawing will be common. As temperatures will rise and fall rapidly as the moon moves into and out of the planet’s shadow, you can expect some crazy weather (think massive cyclones) as a result.
