Very naive router in php doesn't work - keep getting 404 : the requested url was not found on this server

It's not clear in what order you have attempted to implement these changes and what errors you got and when. (And actually how you have tried to debug this, eg. what have you done to determine what files are being called?) For instance, if nt2.conf was being correctly included then you should not have been getting the message "AH00558 - could not reliably determine the server's fully qualified domain", or did you only add the ServerName directive to nt2.conf later? You will need to have called e2ensite nt2.conf to enable nt2.conf (to effectively get it included), but you only mention this later in your question?

What exactly are you seeing in the 404 response? I'm assuming this is the default Apache response and not the 404 response from your PHP script? It doesn't look like your PHP router script (index.php) is being called at all since the $viewDir variable is missing a slash so the PHP require statement would fail and result in a fatal error (not simply a 404). Do you have PHP debugging set? Are PHP errors set to be output? If PHP errors are not output (as in production) then you would expect a 500 Internal Server Error.

If index.php is not being called then the front-controller pattern in .htaccess is not doing anything. To enable the processing of .htaccess files (in other words, to enable Apache directives to be overridden in .htaccess files) then you must set AllowOverride All (or at least FileInfo) in the appropriate <Directory> container. It cannot be left as None (but you must not change AllowOverride None directives that affect parent directories (the server root!) - in the main server config.) If you are using mod_rewrite (RewriteEngine, RewriteCond, RewriteRule, etc.) then you also need to enable mod_rewrite (sudo a2enmod rewrite - as you mention). Failure to enable mod_rewrite would result in a 500 server error if .htaccess was processed (NB: Don't use <IfModule> containers).

You shouldn't need to change the main apache config file (apache.conf) or userdir.conf. What you are adding to these files should only be added to nt2.conf in sites-available.

However, your nt2.conf file is incomplete. It should contain a complete virtual host. Something like the following:

# nt2.conf
<VirtualHost *:80>
    ServerName nt2.com
    ServerAlias www.nt2.com
    ServerAdmin webmaster@localhost  
    DocumentRoot /var/www/nt2  
    <Directory /var/www/nt2>
        Options FollowSymLinks
        Require all granted
        AllowOverride All
    </Directory>
    # Set as appropriate
    ErrorLog "/var/www/nt2/logs/nt2.com-error_log"
    TransferLog "/var/www/nt2/logs/nt2.com-access_log"
</VirtualHost>

The ServerName (and ServerAlias) directive(s) are what Apache is looking for in the request. So, in the above, it is looking for http://nt2.com/<anything> (or www.nt2.com - the alias). By requesting nt2.com this Virtual Host applies. However, you say that nt2.com is not a real domain, so for you to be able to access this locally, you would need to configure your local HOSTS file (or local DNS server) to point this hostname(s) to your local IP address (127.0.0.1 - if all on the same machine or whatever your local IP address is on your LAN). eg. in your local HOSTS file, something like:

127.0.0.1 nt2.com
127.0.0.1 www.nt2.com

(The ServerAlias is entirely optional. For a real domain, this is often used to configure the domain apex and www subdomain.)

However, in your example, you are using localhost as the hostname. This could potentially end up with conflicts with the main config, which maybe why you are having to change the main Apache config. You could use ServerName localhost in the above Virtual Host instead (and ensure there are no conflicts elsewhere) but this is limiting as you would only be able to enable one site at once. (If you want to change localhost and modify this for every site you develop then you probably would just modify the main server config every time and not bother with Virtual Hosts - but that is kinda missing the point.)

You don't need to use .htaccess. You could leave AllowOverride None set and include the contents of the .htaccess file directly inside the <Directory /var/www/nt2> container. But, every change to nt2.conf will require the Apache webserver to be restarted. (Changes to .htaccess do not require a server restart.)

I briefly mentioned above about the error in your router script (index.php)...

$viewDir = __DIR__ . '/views';   switch ($request) {  
case '':  
case '/':  
    require  $viewDir . 'home.php';  
    break;  
case '/users':  
    require  . 'users.php';  
    break;  
case '/contact':  
    require $viewDir . 'contact.php';  
    break;

Note that the $viewDir variable does not end in a slash, so the string concatenation later would result in require <dir>/viewscontact.php - which is going to fail (fatal error).

And the require . 'users.php'; line is an outright error. (?)

RewriteEngine On  
RewriteCond %{REQUEST_FILENAME} !-f  
RewriteCond %{REQUEST_FILENAME} !-d  
RewriteRule  . /index.php [L]

This is "ok", except for the erroneous "formatting" backslashes in your question.

However, you do need to ensure that DirectoryIndex is set correctly, otherwise the PHP router script (index.php) will not be called for requests to the document root (the above mod_rewrite rule does not do this). This might already be set in the main Apache config, however, this is not set by default on Apache. In other words, you should include the following at the top of the .htaccess file:

DirectoryIndex index.php

Minor point, but the slash prefix on the RewriteRule substitution string is not required here and best avoided. For example:

RewriteRule  . index.php [L]

This ensures the rewrite is to a file-path relative to the directory that contains the .htaccess file. /index.php is a URL-path (starting with a slash) that results in the request being remapped to the filesystem.

As an optimisation you can also prevent an unnecessary re-evaluation of the rewritten request by including the following rule immediately after the RewriteEngine On directive:

RewriteRule ^index\.php$ - [L]