If you need BIND9 plugin to work, install the BIND9 application itself (I assume that you're on CentOS: yum install bind, or install bind from the Packages page). Then restart Ajenti and plugin should be available.

These warnings are perfectly normal, as not all plugins are needed for all systems.

Yes, if you need extra PHP packages, just install them from the Packages section (e.g.: php5-mysql) and re-apply your website settings.

А что в логах с другими получателями?

Яндекс почему-то не принимает
"host mx.yandex.ru [213.180.204.89]: 451 4.7.1 Sorry, the service is currently unavailable. Please come back later." >> https://toster.ru/q/38981

Можно проверить, сразу ли успешно отправляются письма, смотря в /var/log/exim/mainlog во время отправки письма, или отправить его используя exim -v recipient@gmail.com.

How is your package repository related to restricting users? If you're on Debian/Ubuntu, check your sources.list file.

Сервер требует аутентификации, пример: http://php.net/manual/en/function.mail.php#39305

It would work in the previous version too, but only after an Ajenti restart