When I create an item inside a chest using a command block, specifically:
/blockdata ~ ~4 ~3 {Items:[{id:diamond_axe}]}
the item appears in the chest.
However, when I hold that item and right click, it disappears. When I left click nothing happens, and when I attack mobs, it doesn't deal any extra damage. It is as if the item is not existing.
That's because it doesn't necessarily exist. The "Count" tag, if not defined, defaults to 0, meaning you have 0 of that item in your inventory. You need to apply the "Count" tag of 1 or higher: