What are the nature of courts martial? Do they belong to the judicial branch of the government?

Courts martial are agencies of executive character, and one of the authorities "for the ordering of courts martial has been held to be attached to the constitutional functions of the President as Commander in Chief, independently of legislation." Unlike courts of law, they are not a portion of the judiciary. 

"Not belonging to the judicial branch of the government, it follows that courts-martial must pertain to the executive department; and they are in fact simply instrumentalities of the executive power, provided by Congress for the President as Commander in Chief, to aid him in properly commanding the army and navy and enforcing discipline therein, and utilized under his orders or those of his authorized military representatives." "It must never be lost sight of that the only legitimate object of military tribunals is to aid the Crown to maintain the discipline and government of the Army. (Ruffy vs. Chief of Staff, G.R. No. L-533, August 20, 1946)

Can military tribunals try civilians when civil courts are open and functioning?

Military tribunals cannot try civilians when civil courts are open and functioning. (Olaguer vs. Military Commission, G.R. No. L-54558, May 22, 1987)

Are the members of the PNP within the jurisdiction of a military court?

Pursuant to R.A. 6975, members of the Philippine National Police are not within the jurisdiction of a military court. Criminal cases involving PNP members shall be within the exclusive jurisdiction of the regular courts. (Quilona vs. The General Court Martial, G.R. No. 96607, March 4, 1992)

In enacting R.A. 7055, did the lawmakers intend to divest the military court's jurisdiction over cases mandated by the Articles of War?

In enacting Rep. Act No. 7055, the lawmakers merely intended to return to the civilian courts the jurisdiction over those offenses that have been traditionally within their jurisdiction, but did not divest the military court's jurisdiction over cases mandated by the Articles of War. Thus, the RTC cannot divest the General Court-Martial of its jurisdiction over those charged with violations of Articles 63 (Disrespect Toward the President etc.), 64 (Disrespect Toward Superior Officer), 67 (Mutiny or Sedition), 96 (Conduct Unbecoming an Officer and a Gentleman) and 97 (General Article) of the Articles of War, as these are specifically included as service-connected offenses or crimes under Section 1 , RA 7055. Pursuant to the same provision of law, the military courts have jurisdiction over these crimes or offenses. (Navales vs. Abaya, G.R. No. 162318, October 25, 2004)

May the court martial could still assume jurisdiction over a military officer who had been compulsorily retired from service?

An officer whose name was dropped from the roll of officers cannot be considered to be outside the jurisdiction of military authorities when military justice proceedings were initiated against him before the termination of his service. Once jurisdiction has been acquired over the officer, it continues until his case is terminated. Military jurisdiction has fully attached to Gen. Gudani inasmuch as both the acts complained of and the initiation of the proceedings against him occurred before he compulsorily retired on 4 October 2005. (Gudani vs. Senga, GR No. 170165, August 15, 2006)