What extensions and names can I use for my homepage?

You may use any of the following for a homepage of a directory:
index.htm , index.html , index.php , index.phtml , index.shtml , index.php3 , default.htm , default.html
default.phtml , default.php , default.php3

If there is not one here that you want to use please use the file .htaccess in public_html to set your own custom default e.g. to set myfile.html as default index file, this should go in .htaccess file

DirectoryIndex myfile.html