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
In Magento 2 Store Views are different variations of one store views in different languages. So in order for store switcher to appear ( it is a natural Magento feature), you need to create different store views for one store.
And in case you have the multi-language website, Magento 2 GeoIP Switcher extension might come in handy. It helps you to direct customers to the store view in their local language based on their location and browser language as well as display prices in customers' local currencies based on their location.
If you want to make sure the extension works as supposed to you can simulate geolocation and try it out on the storefront.
And here is the article on how to create the multi-language website with the extension recommendations in case you just start your way in the new level business: