Yes, the geocoded locations should be written to the cache file. Make sure you have read/write permissions set for this file as well as its containing directory. The cache file should be written to if you set up your pages on the live server the first time the map is accessed.
Yes, for each map a new file will be created. If you remove the particular map from the page, you will not need those files any longer.
Doing testing on your local server just to get to know how the extension is fine, but you will have to get a new api key for when you post your pages onto the remote (live) server.