Configuration > Advanced > Backup entire database downloads the database of your whole survey site as one .sql file: surveys, responses, participant lists, administrators and settings. The file is meant for a LimeSurvey installation on your own server. Files you uploaded, such as images, are not in it.
To keep or move single surveys, export them instead: a survey structure (.lss) for the questions and settings, a survey archive (.lsa) of an active survey for everything including responses and participants. See Export a survey.
Global settings > Storage sets a size limit in megabytes for this direct download.
No. There is no restore in the administration. Contact LimeSurvey support if you need data restored.
Export it as a survey archive and import it on the other site. See Export a survey and Import a survey.