One　of　the　fundamental　problems　in　Finsler　geometry　is　to　study　Finsler　metrics　of　constant　(or　scalar)　curvature.　In　this　paper,　we　refine　and　improve　Chen-Shen-Zhao　equations　characterizing　Finsler　warped　product　metrics　of　scalar　flag　curvature.　In　particular,　we　find　equations　that　characterize　Finsler　warped　product　metrics　of　constant　flag　curvature.　Then　we　improve　the　Chen-Shen-Zhao　result　on　characterizing　Einstein　Finsler　warped　product　metrics.　As　its　application　we　construct　explicitly　many　new　warped　product　Douglas　metrics　of　constant　Ricci　curvature　by　using　known　locally　projectively　flat　spherically　symmetric　metrics　of　constant　flag　curvature.