If I terraformed a moon to have enough artificial gravity/atmosphere to support life, would they end up just crashing into each other? Is there a way to not have this happen, e.g., increasing distance between the two?
This is a partial answer, addressing the atmosphere part of the question only.
The Moon weighs 7.342 × 1022 kg. Earth's atmosphere weighs 5.1480 × 1018 kg. Earth's biosphere (i.e. from 60 km above to 5 km below the surface) weighs between 1 and 4 × 1015 kg. The surface of the Moon is 0.074 that of Earth. So the Moon's biosphere plus atmosphere would approximately weigh:
(Earth's biosphere + Earth's atmosphere) × 0.074 = (4 × 1015 + 5.1480 × 1018) × 0.074 kg = 3.81248 × 1017 kg
biosphere and atmosphere of the Moon divided by the weight of the Moon = (3.81248 × 1017) / (7.342 × 1022) = 0.0000051927 % of the weight of the Moon
which is too small a change from the current weight to make much of a difference on Earth.
As we understand physics today, the only way to increase the gravity of a moon is to increase the mass significantly. Smashing that much stuff into a moon is just not practical, and if it were there would be debris hitting the planet.
If it were possible to add mass "gently" then there should also be the technology to turn the planet-and-moon pair into a binary-planet-pair. But that's so far beyond science as we know it that speculation about the side effects is pointless.
With the density of the moon being 3.34 g/cm3, if you replaced ~80% of the core with osmium, 22.59 g/cm3, it would have the same mass as the Earth, with the diameter of the moon. May need to decrease the mass a little to account for the difference of distance or the radii.
The 2 bodies orbits would have to be altered to share a common barycenter. However, the pull of the sun on the binary planets may cause one of the two to either be ejected or crash into one another.