No, having the MySQLi extension installed will not effect the other extensions or existing pages.
the MySQLi library is available is php version 5.2 and above.
The php version depends on the host. they may be abler to configure the PHP version on a site by site basis, or not for your dedicated hosting, you would need to ask them about it. but PHP v6 is not needed for you to use the MySQLi extension (PHP v6 is not released yet anyway). I ony mentioned it as a reference for when the MySQL library would no longer be available.
Current PHP versions have both the MySQL and MySQLi libraries available.