For the store drop down menu to work on my homepage, do I have to create a new website for each of the stores? Or can I have all of the stores in one website like I have at the moment, which doesn't seem to be showing the drop down menu.
Let's just go ahead and ackowledge that the terminolgy for the scope hierarchy is super confusing. Sorry about that; we've fixed it for Magento 2.
First: Ensure that the additional stores you've created are Enabled.
That said, the hierarchy for Magento scopes in Magento 1 is as follows:
The out-of-box behavior for the store selector is to show the Store Views from the current Store; if your Store Views are currently under the same Store then they should show.
When this problem is fixed you may need GeoIP Store and Currency Switcher to auto redirect your visitors to their local store