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You do not have to. You can simply upload the extracted, cleaned files to your site and access the installation script as http://www.example.com/installation/index.php where www.example.com must be substituted with the domain name of your own site. Kickstart is not a requirement to restore your site. Kickstart merely extracts the archive, just like Akeeba eXtract Wizard.

Please note that since you removed hacked files, it is prudent to remove your site's files before uploading the cleaned ones.

You can restore them on any server, not just the one you backed up from. That's what Akeeba Backup was designed to do. The backup part (being able to restore on the same server) is just a subcase of what Akeeba Backup can do.

If you get server errors, I'd recommend trying to remove or rename your .htaccess file. If that doesn't help, make sure that the target server runs at least PHP 5.2.7 (my software can't work reliably any more with versions of PHP older than six years). If this still doesn't help, take a look at your error log. Also make sure you are not trying to run kickstart.php; once you copy the extracted files, you should not run kickstart.php, you are supposed to run the installation/index.php script.