Geometries, energies, vibrational frequencies, and magnetic properties have been computed at the B3LYP level with the 6-31G* and 6-311+G* basis sets for a family of endohedral closo-boranes, -alanes, and -gallanes Ng@A(12)H(12)(2-) with noble gas atoms (Ng) located in the centers of icosahedral [B-12], [Al-12], and [Ga-12] clusters. The endohedral structures of most of the systems are minima lying above separated Ng + A(12)H(12)(2-) by 166 (He@B12H122-) and 403 (Ne@B12H122-) kcal/mol for boranes: 29 (He@Al12H122-), 63 (Ne@Al12H122-), 154 (Ar@Al12H122-), and 189 (Kr@Al12H122-) kcal/mol for alanes; and 39 (He@Ga12H122-), 71 (Ne@Ga12H122-), and 213 (Ar@Ga12H122-) kcal/mol for gallanes. Three types of transition states are found for the exit of Ng from a cage: via an edge (TS-1), through a face (TS-2), and via a more extensive deformation through a pentagonal cage "neck" (TS-3). The most favorable exit path depends on the rigidity of the cage, the exothermicity of the dissociation, and the relationship between the size of the internal cavity of the cage and the Ng atomic radius. Ng exit via TS-3 is preferred for He@Al12H122-, Ne@AI(12)H(12)(2-), He@Ga12H122-, Ne@Ga12H122-, Ar@AI(12)H(12)(2-), and Kr@Al12H122-. Helium exits via a cage edge (TS-1) for He@B12H122- while for Ne@B12H122- the neon exits via a triangular face (TS-2). Exit barriers (H-exit(+)) are high enough 30-60 kcal/mol) for all helium clusters and for Ne@Al12H122- and Ne@Ga12H122- to ensure the kinetic stability of these systems. The barriers for Ar@AI(12)H(11)(2-) and Kr@Al12H122- decrease to 10-15 kcal/mol, while Ne@B12H122- has a very low exit barrier and is not expected to be stable kinetically. There is a linear dependence of Ng@A(12)H(12)(2-) cage size on the Ng atomic radii; that is, the heavier Ng atoms "bulge" the cages. Nucleus independent chemical shifts (NICS) indicate that all three A(12)H(12)(2-) anions are aromatic but the alanes are the least so. A face- or edge-coordinated external Li+ cation has a moderate effect on the structure and vibrational and magnetic properties of the helium-containing clusters, i.e., Li[Li[He@A(12)H(12)](-). In contrast, for systems with very large exothermicities of Ng exit, Li+ complexation promotes their dissociation. Thus, the internal atom Ne exits from the cage of Li[Ne@B12H12](-)[ and the salt dissociates into Ne + LiB12H12- without barrier. Systems with two Li+ ions located initially above opposite cage faces (Li-2[Ng@A(12)H(12)]) undergo complex intramolecular rearrangements leading to destruction of the icosahedral closo structures.