I think it would be better to say that an aromatic compound is more stable than we would expect it to be on the basis of a conventional structure (drawn with conventional single and double bonds). Thus benzene is less reactive than we would expect for 1,3,5-cyclohexatriene, if it really had alternating single and double bonds. It doesn't undergo characteristic double bond reactions like addition of Br2, and although it can be hydrogenated, the heat of hydrogenation is less exothermic than we would expect for 3 normal double bonds.
It is not, however, more stable than a "non-aromatic" compound like cyclohexane. You have to be careful with your comparisons.
On the other hand, cyclobutadiene and cyclooctatetraene are not aromatic, and behave more like alkenes than arenes. You coukld say they are less stable than benzene.
The lone pair in phenyl anion, and the extra pi bond in benzyne, are orthogonal to the delocalised pi orbitals and to a first approximation do not affect their aromaticity.