Restart nginx:

• sudo systemctl restart nginx - CentOS 7, ArchLinux
• sudo service nginx restart - Ubuntu, Centos 6.x, Debian

If that doesn't work, check your configuration - If you have domain properly configured etc

I looked into your issue a little bit.
The way ajenti website creation works, it isn't possible. When you create a new page, you enter website name only (identificator), not server_name (domain), that is "example.com" by default. I could tell you, how to set server_name to website name and then add www to it automatically, though.

I have a solution for your suggestion number one:

1. Find file "api.py" (path /var/lib/ajenti/plugins/vh/api.py on my CentOS 6 system)
2. Edit line 41, replace '/srv/new-website' with '/var/'+j['name'], so the whole line should look like this: self.root = j.get('root', '/var/'+j['name']) instead. (you can also modify it to your own needs. you can add +'/public_html' at the end of bracket, like this '/var/'+j['name']+'/public_html')
3. Save file and restart Ajenti.

This will change the default template directory. Just beware that after an update package manager will most likely replace this file, so you would have to do it again.
I also highly recommend you to backup the original file in case you made a mistake like missing ', so you can revert it back.

Solution for suggestion number two is to edit the same file as above, and just 2 lines below we have
self.custom_conf = j.get('custom_conf', '').
Now, we can insert our custom-default conf here.
self.custom_conf = j.get('custom_conf', 'include /etc/nginx.templates.d/generic_vhost.conf;')

Yeah, completely agree. I think that point 2 needs an extension though: For every file in /etc/nginx.templates.d/*.conf make a checkbox with filename as checkbox name. IE: I have files /etc/nginx.templates.d/wordpress.conf and /etc/nginx.templates.d/drupal.conf. Ajenti should create checkboxes "Wordpress" and "Drupal" (capitalize first letter).