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Akeeba Backup is only compatible with MySQL.

Akeeba Backup is only compatible with MySQL. This already answers your questions. I don't know how else to say it. Wanna use Akeeba Backup? You must use it with MySQL.

FYI: no, Joomla! does not support SQlite. Currently, Joomla! only supports MySQL, SQL Server and PostgreSQL. Both SQL Server and PostgreSQL do not provide a way to dump the DDL of entities (e.g. the CREATE TABLE SQL statements), making backup problematic. Even though we have implemented an experimental method to dump SQL Server databases, the limitations of what is possible are so many that it makes no sense to push that feature in production. As a result we will only ever support using Akeeba Backup for backing up or restoring your site with the MySQL database server only.

FYI #2: SQLite is approximately 100x SLOWER than MySQL, especially when dealing with concurrent requests, due to the atomicity of the database commits. It's OK for single user environments, such as persisting the per-user settings in a browser or as a temporary cache in multi-tier applications, but it's outright horrendous for practical use on websites.

FYI #3: The majority of Joomla! extensions only support MySQL and compatible database servers such as MariaDB and Percona. There are quite a few reasons for that which would require a book to list. Simply put, if you want to use anything else besides core Joomla! you must use MySQL or your chances for success are slim to non-existent.

FYI #4: The support for database servers beside MySQL was added to Joomla! not with the intention of necessarily supporting the installation of the CMS on such a database server but as a useful feature for enterprise and power users who want to integrate their sites with third party systems running on such database servers. As a result the support for other database server technologies (that the majority of the Joomla! developers, core and third party, has little to no experience with) is limited at best.

Bottom line: stop trying to shoot yourself in the feet and use MySQL or a compatible database server :)

Eek! The host tried to convince you that a server-less on-disk database without caching and atomic commits is faster than a proper relational database server?!?!?! Run! You need to run to a proper host where they know what they're doing. If you're looking for a host I recommend SiteGround, CloudAccess.net or Rochen. I've tried all three and they are excellent (and they do specialise in Joomla! hosting which is a big plus).