Your domain has no server address records: http://www.dnsstuff.com/tools#dnsReport|type=domain&&value=mailingdove.com

You need to check the actual error that your browser is reporting. Looks like a network or DNS problem so far.

Здравствуйте. Может, но только на уровне редактирования конфигов через панель. Для полноценного авто-вебхостинга - только NGINX+FPM

Make sure that all your Ajenti V packages are up to date, this was an issue in an old version.
Also, your web-related services (php5-fpm, nginx etc) might be disabled in autostart (use chkconfig on).

Yes, you should keep it running, but you need to allow nginx process to raise its limit values (unfortunately I have no experience with SELinux)

In fact I'm running out of ideas already :(

SELinux might affect this, is it enabled?

Try to raise the limit manually for the www-data user: http://www.cyberciti.biz/faq/linux-increase-the-ma...
Restart NGINX and see if rlimit message still appears in the log.

Are you running an actual dedicated/VPS server or under some kind of OS-leve virtualization like OpenVZ? The problem is that www-data user isn't allowed to raise RLIMIT_NOFILE, but normally it should be.

1.Domain name for phpMyAdmin is incorrect. Neither 192.168.2.5 nor 'phpMyAdmin' are domains. Actually you *must* set up some DNS entries to discern between two websites on your server. 
2. Have you manually lowered the number-of-files limits on the system? See: http://forum.nginx.org/read.php?2,248426,248427