That's really strange, everything seems correct in your config. Check /var/log/nginx/<website>.error.log for both domains after you see 403 error.

There's no way adding a domain can break permissions. In any case, there's a "Fix permissions" button on the website page, he might want to try that. It looks more like there's somehow some 'static files' content taking precedence, without directory index allowed.

Сейчас попробовал установить триал 10.1 с их сайта - все встало сразу. А у вас какая версия?

Please post your /etc/ajenti/vh.json file with subdomain enabled. (be careful to remove passwords)

Grants permissions from all website users to all website DBs

Doesn't happen to me. Please post /etc/ajenti/vh.json.

This means repository was recently updated. Unable to fetch some archives, maybe run apt-get update

Not only forwarding, but also simple outgoing mail exchange both require outbound SMTP connectivity.

Yes, totally. SMTP is obviously required to send e-mails (and it's not only used for connection between e-mail client and sender e-mail server, but also between the sender e-mail server and destination e-mail server as well).

Try to actually send it by running exim -v address@gmail.com
and inputting:

Subject: test
From: root@localhost

test
.
<newline>