That's fine, though cold feed, while it adds hydration, can warm them up.
The body doesn't like cold things, so it works hard to warm them up for digestion.
Wet feed is fine.
It actually can. You don't want them to fill up on high carb/fat low nutrition "store treats" or high fiber/water fruits and vegetables.
It's best not to over do melon, while it has electrolytes, it's also high in fiber and water which can cause diarrhea, which will dehydrate your birds.