Greece was never occupied by British troops or any form of administration. The only Empire forces ever in Greece were those sent to fight the Germans in 41.
Australian and NZ forces were not being moved away to the Middle East when Japan entered the war. That was when the 2nd AIF returned home. The 2nd New Zealand Division remained in the ETO, as they were not required for home defence or nearby campaigns; the 3rd was always in the Pacific, and the 1st, 4th and 5th were on home defence duties.
The Australian and NZ expeditionary forces had been sent from early 1940, when Japan was not at war.
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for the most part agree,
the first greek leader metaxas did not want british aid, (due to unpleasant memories of being pushed around by british), but allowed british air force squadrons stationed in greece.
when he died in jan 1941 his successor allowed a british army (primarily NZ and AUS divisions) to man the bulgarian border (freeing up greek troops to fight italy on albanian front). This was before germans attacked in april. After retreat, 30,000 british troops and only 10,000 greek moved to crete with brits taking overall command.
Iran was trying to nationalize their resources (oil) as they had been exploited by the british, as you pointed out they were effectively invaded and taken over by ussr/britain.
Iraq had regained their independence from britain in early 1930s but Britain retained the right to station troops there and exploit their oil. (Persia of course was independent)
Empire essentially "occupied" egypt for 60 yrs to control oil and suez canal.
As for NZ they requested their armed forces fighting italy and germany to return home following japan entering the war, britain was reluctant to do so and finally as japan got closer brit allowed the nz navy to go back but their airmen and 2nd division would remain under british control. essentially the result of these frustrations would result in kiwis no longer thinking of themselves as british.
But by and large this is what commonwealth/dependents/etc are for, to be used and for use of the country running the empire.
Just seems highly "unfair" (couldn't think of another term) that german satelites who have infantry on the frontline have horrendous stats whereas british CW troops have tremendous stats.