Colour schemes of WW2 aircraft were generally functional, even the whackier schemes had a purpose, whether for unit identification or specific functions. There was often a clash between functions, for example early in the war RAF fighter planes were camouflaged but then had to add bright yellow bands to their roundels to aid identification and reduce confusion.

The overall grey colour of early Zeros with bright rising sun markings shows the same tension. The grey was similar to current low visibility military schemes, very handy in dogfighting where seeing the enemy before he sees you is vital, but without IFF technology pilots still needed to rapidly know who was who, hence the bright markings.

Incidentally the colour was not meant to be white as you suggested. Factory paint was either a glossy amber grey or olive grey but it oxidised to a paler colour in use, as shown here. It seems that modern restorations and replicas may be painting their planes this much paler colour from scratch.

Grey Zeros based on land tended to have their upper surfaces painted in the field with a variety of green and browns schemes, often quite roughly. They preferred not to be visible when parked on airfields.

From 1943 factory fresh Zeros did have the dark green upper surfaces that you describe. I don't know why, but would point out that US Navy carrier planes progressed from light grey, through medium shades to dark blue during the war so there may have been operational reasons why darker colours were preferred by both sides later on. Anything beyond that would be me speculating.

Both the Japanese Army and Navy operated Zeroes. As far as I am aware, the Japanese Navy used a light-coloured camouflage while the Japanese Army used a green camouflage.