Ooh, fun question, or in fact, a bunch of questions bundled together
##Endowing a moon with an atmosphere.
It is still unclear what the mechanisms of atmospheric accretion and retention are exactly, since you have moons like Titan (at 1.3E+23 kg it is 0.0225 $M_{⊕}$ --Earth masses-- or about 1.8 times the size of our moon) with a dense atmosphere (146.7 kPa or about 140% of Earth's atmosphere), and heavier planets like Mercury (0.055 $M_{⊕}$ but only trace atmosphere) and Mars (0.107 $M_{⊕}$ with 0.636 kPa or ~1% of Earth's). Moreover, even though Earth and Venus are similar in size (Earth is slightly larger), Venus has an atmosphere ~90 times denser.
So obviously the moon's mass, distance from primary star, original gas endowment, and presence and strength of a magnetosphere play a role:
Higher mass means higher escape velocity, helping the object retain the gases. As a rule of thumb, you generally want the object's escape velocity to be at least 6 times the particular gas's mean velocity. For example, Earth has an escape velocity of 11.2 km/s, while O2 at 20 C has an average molecular speed of 0.48 km/s, so Earth can retain molecular oxygen for billions of years. Mars has an escape velocity of 5 km/s, so molecular hydrogen --average speed at 20C being 1.9 km/s-- cannot be retained long term on either planet.
Distance from primary star has 2 effects. The star's radiation warms the gas on the moon/planet surface (increasing molecular speed), and the star emits stelar winds, a flow of charged particles which act to strip planets of their gas (hence Mercury, closest to the sun, having only trace amounts of gas)
Original gas endowment at the end of the planetary formation stage of the first ~200 million years of the star system's existence. This is hardest to factor in, but might account for why Venus and Earth are so different (that and the oceans on Earth). For instance, it could be that the moon-creating impact dispersed some/most of Earth's initial atmosphere.
Magnetospheres deflect incoming solar winds, in a sense protecting the atmosphere from impacts with the highly charged particles of the solar wind and the molecular dissociation that the uppermost layers of the atmosphere would experience otherwise. These are also mysterious. For terrestrial-sized planets, they seem linked with activity in the molten iron core and speed of rotation, but we don't truly understand the mechanics well yet.
##So, what can we conclude?
Well, it depends on what kind of atmosphere you want. If you're ok with Titan (recall: its mass is 1.8 Earth moon's, ~2% of Earth) you can get 98% Nitrogen and some methane. Not breathable, but you could live with sealed scuba-like-gear if it were warm enough, not the bulky Moon-space suits of the Apollo missions. Surface gravity would be similar to the moon (more mass but less dense). You'd have lakes of gasoline-like compounds on the surface. Tides would be similar to what you see now, while the moon would be larger in the planet's sky due to its atmosphere. Of course, this might be hard to explain in terms of formation and how its maintained, since bringing Titan to Earth-orbit would make it 10 times warmer and would probably results in full atmospheric loss in about 10 million years, but weirder things have happened.
If you want an Earth-like atmosphere with water and all that, stable over the billions of years, you may need a molten core for magnetosphere, high enough gravity, etc. which means you probably need something about 0.4 $M_{⊕}$ or larger, so a moon larger than Mars. At that point, it's not really a moon, but a double planet. That comes with its own headaches. With mass that high, the two worlds would quickly become tidally locked, which is to say rotate in synchrony, and always show the same face to the other. (Your 'moon' would only ever be visible from half the planet, no tides at all and possibly very long days).
