What restrictions apply in international airspace?
When a US registered pilot is operating in international airspace, he is bound by both the US aviation regulations (14 CFR) and ICAO Annex 2, as required by 14 CFR 91.703.  It also requires compliance with MNPS and RVSM airspace rules where appropriate.


  Sec. 91.703 - Operations of civil aircraft of U.S. registry outside of the United States.
  
  (a) Each person operating a civil aircraft of U.S. registry outside of
  the United States shall--
  (1) When over the high seas, comply with
  annex 2 (Rules of the Air) to the Convention on International Civil
  Aviation and with Secs. 91.117(c), 91.127, 91.129, and 91.131;
  (2) When within a foreign country, comply with the regulations relating to
  the flight and maneuver of aircraft there in force;
  [(3) Except for
  Sec. Sec. 91.117(a), 91.307(b), 91.309, 91.323, and 91.711, comply
  with this part so far as it is not inconsistent with applicable
  regulations of the foreign country where the aircraft is operated or
  annex 2 of the Convention on International Civil Aviation; and]
  (4) When operating within airspace designated as Minimum Navigation
  Performance Specifications (MNPS) airspace, comply with Sec. 91.705.
  When operating within airspace designated as Reduced Vertical
  Separation Minimum (RVSM) airspace, comply with Sec. 91.706.  
  
  (b) Annex
  2 to the Convention on International Civil Aviation, Ninth
  Edition--July 1990, with Amendments through Amendment 32 effective
  February 19, 1996, to which reference is made in this part, is
  incorporated into this part and made a part hereof as provided in 5
  U.S.C. Sec. 552 and pursuant to 1 CFR part 51. Annex 2 (including a
  complete historic file of changes thereto) is available for public
  inspection at the Rules Docket, AGC-200, Federal Aviation
  Administration, 800 Independence Avenue SW., Washington, DC 20591; or
  at the Office of the Federal Register, 800 North Capitol Street, NW.,
  Suite 700, Washington, DC. In addition, Annex 2 may be purchased from
  the International Civil Aviation Organization (Attention: Distribution
  Officer), P.O. Box 400, Succursale, Place de L'Aviation
  Internationale, 1000 Sherbrooke Street West, Montreal, Quebec, Canada
  H3A 2R2.


Another unrelated FAA document explains it in plain language:


  International Civil Aviation Organization (ICAO)
  
  ICAO is an agency of the United Nations that originated from the
  signing of the Convention on International Civil Aviation in December
  1944 (referred to as the Chicago Convention). This organization was
  formed to promote the safe and orderly development of international
  civil aviation. The Chicago Convention produced International
  Standards and Recommended Practices (SARP) aimed at standardizing
  international civil aviation operational practices and services.
  Currently, these SARP's are contained in 18 annexes to the Chicago
  Convention. In particular, Annex 2, Rules of the Air, and Annex 11,
  Air Traffic Services, are pertinent to this paper as they relate to
  civil aircraft operations, the establishment of airspace, and air
  traffic control (ATC) services in international airspace. Although not
  addressed in this paper, it should be noted that the SARP's are
  augmented by various ICAO documents containing Procedures for Air
  Navigation Services (PANS) that amplify the basic principles in the
  SARP's. Further, to supplement the worldwide procedures contained in
  the annexes and PANS, ICAO develops Regional Supplementary Procedures,
  which are part of the Air Navigation Plans agreed upon to meet the
  needs of specific geographical areas.
  
  Two articles of the Chicago Convention also have a bearing on the
  subject of this paper:
  
  Article 3:
  
  Exempts state-owned aircraft (which includes military aircraft) from
  the provisions of Annex 11 and its Standards and Recommended
  Practices. (Note: As an ICAO Contracting State, the U.S. has agreed
  that its state aircraft will be operated in international airspace
  with due regard for the safety of civil aircraft.)
  
  Article 12:
  
  Obligates each ICAO Contracting State to adopt measures to insure that
  persons operating an aircraft within its territory shall comply with
  that state's air traffic rules; or with Annex 2, Rules of the Air,
  when operating over the high seas. The U.S. satisfied this
  responsibility through Title 14, Code of Federal Regulations (CFR)
  Part 91 which requires that operators of aircraft comply with U.S.
  operating rules when in the U.S., and that U.S.-registered aircraft
  comply with Annex 2 when over the high seas (see section 91.703).
  However, section 91.703 applies only to civil aircraft, and state
  aircraft operating outside the U.S. are only subject to the "due
  regard" provisions of Article 3 of the Convention.
  
  Under ICAO agreements, the SARP's in Annex 11 apply to airspace under
  the jurisdiction of a contracting state which has accepted the
  responsibility of providing air traffic services over the high seas,
  or in airspace of undetermined sovereignty.