Engineering heterostructures with various types of quantum materials can provide an intriguing playground for studying exotic physics induced by proximity effect. Here, we report the successful synthesis of iron-based superconductor FeSe\(_{x}\)Te\(_{1-x}\) (FST) thin films in the entire composition of \(0 \leq x \leq 1\) and its heterostructure with a magnetic topological insulator by using molecular beam epitaxy. Superconductivity is observed in the FST films with an optimal superconducting transition temperature \(T_c\) \(\sim\) 12 K at around x = 0.1. We found that superconductivity survives in the very Te-rich films (\(x \leq 0.05\)), showing stark contrast to bulk crystals with suppression of superconductivity due to an appearance of bicollinear antiferromagnetism accompanied by monoclinic structural transition. By examining thickness t dependence on electrical transport properties, we observed strong suppression of the structural transition in films below t \(\sim\) 100 nm, suggesting that substrate effects may stabilize superconducting phase near the interface. Furthermore, we fabricated all chalcogenide-based heterointerface between FST and magnetic topological insulator (Cr,Bi,Sb)\(_{2}\)Te\(_{3}\) for the first time, observing both superconductivity and large anomalous Hall conductivity. The anomalous Hall conductivity increases with decreasing temperature, approaching to the quantized value of \(e^2/h\) down to the measurable minimum temperature at \(T_c\). The result suggests coexistence of magnetic and superconducting gaps at low temperatures opening at the top and bottom surfaces, respectively. Our novel magnetic topological insulator/superconductor heterostructure could be an ideal platform to explore chiral Majorana edge mode.