What are the Jewish ethics on war trophies/looting?

I know this a very weird question but honestly it came to me as I was laying in my bed and I thought I would ask real people as I doubt online resources say much about this. Also this is not in relation to nonsense about Aza.

My understanding is that theft is a very severe sin and one shouldn’t steal anything even the smallest of items. However in situations of warfare it’s likely if you’re in a house that the inhabitants fled or are dead. Is it wrong to take an item, even as a “souvenir”? Also what about off a defeated enemy? Would it be wrong to take a patch/helmet or even a note that the man wrote? Even not as a trophy but as something to remember the person you killed.

Finally, I remember hearing something about Jewish kings taking the spoils of war for the nation, I think specifically cattle?

Again sorry for the weird question