Did the Japanese use German aircraft during WWII?

  The Allies mistakenly thought the Germans were exporting their fighters and bombers to Japan, or licensing them to build them in Japan. The Allies assigned them codenames (nicknames) expecting to see them in combat. This was an early war rumor, mainly from bad intelligence and the incorect feeling that the Japanese were incapable of designing their own aircraft (proved wrong with the combat debut of the A6M2 "Zeke / Zero")

JU-87 "Irene"	JU-88 "Janice"	HE-111 "Bess"

Testing and Influence of German Aircraft
   The Japanese did have a few examples of the above aircraft sent to Japan, where they were run through performance tests, or flown in mock combat against their own fighters. The Japanese Army brought a JU-87A "Stuka" to Japan, where it might have influenced the design of wing dive breaks, seen on there their D3A1 Val The Japanese Ju-87 remained in a hanger-museum at Tokorozawa, but was destroyed during an Allied air raid toward the end of the war.

Mistaken Identity - Me-109 and the Ki-61 "Tony"
  The rumor that Allied pilots encountered ME-109's in combat is a case of mistaken identity. When the JAAF's Ki-61 (Tony) was first encountered in April of 1943, it was sometime mistakenly identified as a ME-109.

ME-109 "Mike"	ME-110 "Doc"	FW-190 "Fred"

 Others, thought that the new plane was an Italian export, therefore, it was codenamed "Tony". Instead, the Ki-61 was neither. It was a Japanese design, but the rumors persisted. I have read several personal memories that talk about Navy pilots in the making the identification mistake of ME-109 instead of Ki-61.

German Secret Weapons in Japan
   At the end of the war, the Japanese did test the ME-163 (Japanese designation J8M1) The IJN acquired manufacturing rights to build this rocket fighter. By the end of the war, flight tests were made in glider versions of the plane only.

  The famous ME-262 (In Japan, Nakajima built, and know as "Kikka") The design of the Kikka was based on the Me-262, but it became scaled down because of the loss of some of the blueprints from Germany. One successful & one aborted flight were made before the end of WWII.  The Allies never had codenames for either of these planes because they were not encountered in combat.

German Aircraft Wrecks in New Guinea
  Although the Me-109 never flew in New Guinea, there is a legacy of German aircraft in Papua New Guinea, but it has nothing to do with WWII! Previous to WWI, Germany was the colonial masters of the northern coast of New Guinea - hence names like Finchafen, Alexishafen, Mt. Wilhelm, etc. Their missionary influence is still present to this day.

  During the 1930's those German ex-patriates imported German aircraft to New Guinea. At Alexishafen, there is the remains of a Junkers W 34 and at the Bayier River in the Highlands, the reamains of a Ju-52

Additional Reading
Mikesh, Robert C. "Japanese Aircraft Code Names & Designations" Schiffer Miliary History, PA USA. 1993.