This paper reports measurements of processes: e+e−→γK0SK±π∓, e+e−→γK+K−π0, e+e−→γϕη, and e+e−→γϕπ0. The initial-state radiated photon allows to cover the hadronic final state in the energy range from thresholds up to ≈4.6  GeV. The overall size of the data sample analyzed is 232  fb−1, collected by the BABAR detector running at the PEP-II e+e− storage ring. From the Dalitz plot analysis of the K0SK±π∓ final state, moduli, and relative phase of the isoscalar and the isovector components of the e+e−→KK∗(892) cross section are determined. Parameters of ϕ and ρ recurrences are also measured, using a global fitting procedure which exploits the interconnection among amplitudes, moduli, and phases of the e+e−→K0SK±π∓, K+K−π0, ϕη final states. The cross section for the OZI-forbidden process e+e−→ϕπ0, and the J/ψ branching fractions to KK∗(892) and K+K−η are also measured.