Can the President create a public office such as Philippine Truth Commission, to investigate reported cases of graft and corruption?

No, the President has no power to create a public office. The power to create a public office is not shared by the Congress with the President/Executive. Separation of powers shall be observed. It is already settled that the President’s power of control can only mean the power of an officer to alter, modify, or set aside what a subordinate officer had done in the performance of his duties, and to substitute the judgment of the former for that of the latter\ As such, the creation by the President of a public office like the Truth Commission, without either a provision of the Constitution or a proper law enacted by Congress authorizing such creation, is not an act that the power of control includes. (Biraogo v. The Philippine Truth Commission, G.R. No. 192935, 7 December 2010, Bersamin, J. separate opinion)