the preferred information about location should be stored in a user's profile or the selection is based on selected value of drop down list on a landing page?
After the location selection should the user be redirected to appropriate page? Does it mean that only store information (address, hours) is different the rest of site is the same?
The solution I think about is to create Stores folder and under it 10 documents based on your custom Store document type.
Now to display the correct one on a Home page you could do a simple trick.
On a Home page (or other document where Store details should be displayed dynamically) insert for example a repeater web part. It will display information about store. To this page will the user be redirected after he made a selection in a drop down list (you can create a simple redirection web part). For example if he selects London he will be redirected to http://domain.com/Store/London.aspx
you can create a query parameter from the London part of URL. Then you can use it in the Where condition of a repeater web part to select only this one Store.
I hope the description makes sense.