You could use a plugin like SB Welcome Email Editor to edit the welcome message. You can use HTML here, so adding an image is easy.
To make sure only invited members can see the site untick the Anyone can register setting on wp-admin/options-general.php and install a plugin like Members to require users to log in before they can see anything.
For your other questions … ask new, separate questions. Be aware that questions about browser behavior are off topic here. 😉