Migrating to WPEngine
I have only used CiviCRM at WPE on test sites.   

You will have to share the same DB as WP.  

You need to prepare teh backup for restore offline.   That said, I mean that if backed up from another server, the DEFINER (used in Triggers and Routines) have to be stripped out, and the database should be compressed.   You'll need to restore via phpmyadmin from my.wpengine.com

A word of warning, WP Engine reserves the right to change the database user.   When they do this they will update wp-config.php, but not civicrm.settings.php.   This has not happened to me on CiviCRM (keep in mind I only use WPE for testing on CiviCRM) , but it has happened on WP sites I have supported at WPE.