Why do fully homo-morphic constructions use 'ring' or 'lattice' structures? Is there a significantly advantage to these data structures, or is it simply the status-quo and the easiest to use for describing constructions?
Lattices are used simply because we do not know how to build homomorphic encryption from any other problems, like factoring or discrete logarithm.  Their mathematical structure was found to simultaneously enable homomorphic addition and multiplication while other bases for encryption could handle at most one of them.  (For the case of ideal lattices/rings, this seems intuitive:  If ciphertexts are elements of a ring then you can naturally add and multiply them, although things are not really that simple.  It was later found that the ring structure could be removed, and that any lattice would suffice.)

Advantages like quantum resistance etc are only incidental.