The battle cry of the Italian troops before and during WW1 was actually "Avanti Savoia!" but it was also used throughout WW2, only more sporadically. It fell out of use after the Republic was established in 1946.

did Japanese troops actually shout something along the lines of "FOR THE EMPERORRRR" ?

You're probably thinking of banzai. "Banzai" (万歳) means "ten thousand years," and derives from a Chinese saying of the same meaning - wansui.

Yes, Japanese soldiers did shout banzai. Banzai is a term used by Japanese people today, such as at sporting events.

If one wants to specify the emperor, one would shout Tennouheika Banzai, roughly meaning "ten thousand years to the Emperor."

Jai Mahakali, Ayo Gorkhali' (' Victory to Goddess Mahakali, the Gorkhas are coming'). Battle cry of the Ghurka regiments. I have also seen the simple Ayo Gorkhali related in WW2 memoirs.