The 1-kink skyrmion is a topological spin texture composed of a skyrmion having a chiral kink inside its Néel domain wall, this chiral kink being also called domain wall skyrmion. By micromagnetic simulations we have studied the stability of the 1-kink skyrmion as well as its spin-driven dynamics in Pt/Co/MgO. Such trilayer system allows simultaneously the stabilization of various spin textures including skyrmion (Sk, topological charge Q = 1), 1-kink skyrmion (1kSk, Q = 2), and domain wall skyrmion (DWSk, Q = 1). The 1-kink skyrmion is found to be robust against both thermal fluctuations and normally distributed anisotropy for different grainlike region sizes. Such magnetization pattern has a certain resilience to the variation of the Dzyaloshinskii-Moriya interaction and applied magnetic field. The static analysis is complemented with the investigation of the dynamic regime of motion under electrical current. In the limit of moderate injected current the skyrmion and the 1-kink skyrmion behave similarly. However, depending on the geometry of the injected current, the 1-kink skyrmion can be transformed into a single skyrmion or splits in two skyrmions, opening the path to conceive high density logical devices, enabling novel functionalities for logic operations.