I managed to solve all problems related with setting up different domains, here's how:
1) I applied
- 0755 permissions to ALL directories "/var/www/php-fcgi-scripts/<user>" and file "php-fcgi-starter" at the end of the path
- 0755 permissions to ALL directories "/etc/apache2/sites-available/" and 644 to ALL files at the end of the path
- 0755 permissions to ALL directories "/var/www/<user>/web", 600 for file "/var/www/<user>/php.ini" and 600 to ALL files "/var/www/<user>/web/*"
2) I enabled suexec as I had disabled it by accident
<root># a2enmod suexec
Now, I have to experiment with mass virtual hosting and 3rd level subdomains!