With regards to buying/renting:
1) You rent an item for a nominal fee (50-100 rupees per item), and you keep it until you die.
2) When you die, any rented items will be returned to Ravio. So in theory, if you never die, you never have to give up the items.
3) You cannot upgrade rented items.
1) You can buy an item for a high fee (800-1200 rupees per item), and you thus own it and can never lose it.
2) You can upgrade any items that you've bought to the second level through the Maiamai sidequest.
3) Upgraded items are ridiculously powerful, such as the Nice Fire Rod, so it is in your best interest to buy them so you can upgrade them