Yes, Archival process can be scheduled to run on daily, weekly or monthly basis in background without intervention.
Yes, any standard and custom objects can be archived. In addition, field history, deleted records, audit trail can also be archived.
When your business data grows over time, lots of unused or historical data will occupy storage space. Since storage space is expensive in Salesforce, moving these unused old data to another low cost storage will reduce your storage cost, because these cannot be deleted for audit/analytics reasons.
Backup does not delete source data but rather copies to secondary storage and this will not reduce the Salesforce storage cost. DataArchiva is also capable to work as Backup or in Archive mode.
Yes: Archived data can be pulled using REST/SOAP api.
Since unused/old data is moved to Archive, the application query performance will be improved as it has deal with lesser data.
Every event related to archival is audited, such as when data got archived, scheduled, on error, etc