YES, MOONS CAN - AND DO - EXCHANGE ORBITS
In fact, this is the case for the two small Saturnian moons Janus and Epimetheus. Their orbits are only about 50 km apart; a fraction of their individual diameters of about 120 and 180 kilometers. As the one in lower orbit approaches the other, their mutual gravities make them swich places, so the moon that was outermost becomes innermost.
NO, THEY COULD NOT SHARE AN ATMOSPHERE
In spite of the closeness of the orbits of the two moons, they never come anywhere near enough to share an atmosphere - even if they had one. In fact, they never get closer than about 15,000 km to each other. Even the Earth's atmosphere is only about 100 km thick, far less than this. The lower-orbiting, faster-moving moon never actually catches up to the outer, slower-moving moon; instead, they exchange angular momentum, making the forward moon move into a lower, faster orbit, while the rear moon moves into a higher, slower orbit (see image below). It is kind of like relay runners passing batons to each other.
DIFFERENT ORBITAL CONFIGURATIONS ARE VERY UNLIKELY TO ALLOW REGULARLY SHARING AN ATMOSPHERE
We can imagine very different orbital configurations from the one above, such as the moons having perpendicular orbits or orbits of very different eccentricity. No matter the configuration, if the moons were to move close enough to share an atmosphere, the friction with the atmosphere will slow their relative speeds down and change their orbits. This will likely cause the moons to crash into each other after a number of passages.
More importantly, there is the Roche limit to consider. When two celestial bodies come too close, the tidal forces between them will make one or both disintegrate - their closer parts are accelerated faster than the more distant parts. Our Moon could not get closer than about 9,500 km to the Earth without breaking up. The smaller the two moons in your question are, the lower is the Roche limit - but so, also is the thickness of the atmosphere their gravities could hold. For two moons of roughly equal density, the Roche limit is at least 1.26 times the radius of the smaller moon (and higher if their sizes are more different), meaning that the closest the two moons can get is more than one-quarter the radius of the smaller moon. Saturn's moon Titan has a very thick atmosphere (much thicker than the Earth's) of about 600 km. The Roche limit for two such moons is 670 km, so they could actually get close enough to share an atmosphere, however briefly (their passage will be at a very high relative velocity)- but the tidal forces of their passage will strip their atmospheres off, losing a lot to space. I don't imagine it would take very many passages before their atmospheres thin too much to be shared.
To sum up: In very extreme cases, with two very similarly sized moons with very thick atmospheres, they could just barely pass each other close enough to share atmospheres without breaking up from tidal stresses - though these stresses would be very hard on the moons, causing major earthquakes and massive tidal waves if they have oceans. However, friction and loss of atmosphere would make it impossible for such passages to be regular events.