Short answers, pulled from the story.
The Anglo-Iraqi War was triggered by the Golden Square coup of the 1st of April 1941, in which four senior Iraqi military commanders and nationalist politician Rashid Ali al-Gaylani seized power from the pro-British Regent Prince Abd al-Ilah. Rashid Ali's new "National Defence Government" sought German and Italian military assistance and moved to restrict British treaty rights, prompting Britain to respond with military force.
RAF Habbaniya was the central battleground of the war. On the 2nd of May 1941, Iraqi forces numbering around 9,000 troops surrounded the base with approximately 50 field guns. The garrison of about 2,500 men, including trainee pilots and Assyrian levies, launched pre-emptive airstrikes. By the 6th of May the Iraqi besieging force had withdrawn, abandoning substantial quantities of arms and equipment.
Yes. Germany deployed Fliegerführer Irak, a Luftwaffe force of between 21 and 29 aircraft painted with Royal Iraqi Air Force markings, which operated out of Mosul from mid-May 1941 under Colonel Werner Junck. Italy sent 12 Fiat CR.42 biplane fighters of the 155.a Squadriglia to Mosul on the 27th of May. Vichy France facilitated both efforts by providing Syrian airfields and transit for supply trains carrying rifles, machine guns, field guns, and artillery shells to Iraq.
The Anglo-Iraqi War lasted from the 2nd of May 1941 to the armistice signed on the 31st of May 1941. Battle honours were awarded to 16 units for service in Iraq between the 2nd and the 31st of May 1941.
Rashid Ali fled Iraq on the 29th of May 1941, travelling to Persia and then Turkey, Italy, and finally Berlin, where he was welcomed by Adolf Hitler as the head of an Iraqi government-in-exile. The Grand Mufti of Jerusalem, Amin al-Husseini, fled with him.
The war's outcome restored the pro-British Regent Abd al-Ilah to power and secured Iraq as a land bridge between British forces in Egypt and India. Britain's anger at Vichy France for supplying the Axis through Syria directly led to the Syria-Lebanon campaign of June and July 1941. Forces reorganised as Paiforce also participated in the Anglo-Soviet invasion of Persia in August and September 1941, and later built forward defences against a potential German advance through the Caucasus.