Hi! Sorry, I haven't really been able to reply lately.
You can't drop items, but you can't fill up the inventory either. Once you have more than 5 items the option to "scroll" up and down using the Z and X keys will become visible. If you can't pick something up, it's because it can't be picked up.
The 1-5 on screen represent the quick-key that will select that item in the inventory.