Rare-earth substitution in the multiferroic BiFeO3 (BFO) material holds promise for resolving drawbacks inherent to pure BFO, and for enhancing piezoelectric and magneto-electric properties via a control of structural and magnetic characteristics. Rare-earth-doped BFO solid solutions also exhibit unresolved features, such as the precise nature and atomic characteristics of some intermediate phases. Here, an effective Hamiltonian scheme is developed that allows the investigation of finite-temperature properties of these systems from an atomistic point of view. In addition to reproducing experimental results of Nd-doped BFO on structural and magnetic transitions with temperature and composition, this scheme also provides an answer (in form of nanotwins) to these intermediate phases. A striking magneto-electric effectnamely a paramagnetic-to-antiferromagnetic transition that is induced by an applied electric fieldis further predicted near critical compositions, with the resulting structural path being dependent on the orientation of the electric field relative to the antiferroelectric vector.