How do I find out where the web site's code resides?

First of all you should check what websites are hosted on the server

# apachectl -t -D DUMP_VHOSTS

Then when you will find a site check corresponding configuration file for the option DocumentRoot. For example

# apachectl -t -D DUMP_VHOSTS
VirtualHost configuration:
wildcard NameVirtualHosts and _default_ servers:
*:80                   is a NameVirtualHost
         default server 192.168.88.87 (/etc/httpd/conf.d/192.168.88.87.conf:1)
         port 80 namevhost 192.168.88.87 (/etc/httpd/conf.d/192.168.88.87.conf:1)
         port 80 namevhost gl-hooks.example.net (/etc/httpd/conf.d/hooks.conf:1)
                 alias example.net
                 alias www.example.net

You want to know where is resides a website example.net

# grep DocumentRoot /etc/httpd/conf.d/hooks.conf
    DocumentRoot /vhosts/gl-hooks.example.net/

# cd /vhosts/gl-hooks.example.net/
# ls -la
total 4484
drwxr-xr-x  6 apache apache    4096 Feb 10 11:59 .
drwxr-xr-x 14 root   root      4096 Feb 23 08:54 ..
-rw-r--r--  1 root   root      1078 Dec 19 09:31 favicon.ico
-rw-r--r--  1 apache apache     195 Dec 25 14:51 .htaccess
-rw-r--r--  1 apache apache      98 Dec  7 10:52 index.html

Should also be on the lookout for aliases and redirects/rewrites

You also should paid attention on any alias directives. For example with the following settings

<VirtualHost *:80>
   ServerName example.net
   ServerAlias www.example.net
   ...
   DocumentRoot /vhosts/default/public_html/
   Alias /api/ /vhosts/default/public_api/
   ...
</VirtualHost>

When you will access http://example.net/some.file.html - apache will look the file at /vhosts/default/public_html/, at the same time with http://example.net/api/some.file.html the file will be looked at /vhosts/default/public_api/.

What about rewrites/redirects, especially programmatic (when redirects are triggered by some php code), I think there is no easy way to find such cases.

Try using find

find / -type f \( -iname "*index.html*" -o -iname "*index.php*" \) 2> /dev/null

Otherwise assuming Apache has been installed from Ubuntu repositories, look in /etc/apache2/sites-available, i.e.

grep -niR "thedomainname" /etc/apache2/sites-available

If the website has an apache VHOST defined, that might locate the config file, then look in that file for "documentroot" this should tell you the location of the source code

Another method, which can be useful for debugging a website (or any process for that matter) is to use lsof (which may not be on path, commonly found in /sbin/lsof)

lsof -s [PID] will list all the files the given process has a handle on, and can be useful to see exactly what is being used (this includes your html/php files, as well as log files and libraries the site needs)

I have no idea where to ... find the index.html/index.php which gets loaded.

Look for page source files

One approach is to browse the site to find a more unique page - lets say newcontactform.php - ideally one that is unlikely to appear in other sites hosted by the same server.

You can then try

locate newcontactform.php

if that fails, follow by

find / -name newcontactform.php

this should produce a managably small list of candidates.

You can then inspect the files, do diffs and if necessary try small changes (e.g. insert an HTML comment) to verify that the file indeed produces the page.

Find the configs

Sometimes config files are evident in the output of the ps command. Worst case is ps -ef | grep -e 'apache|httpd' but more creative use of ps options might be worth exploring.

You can look for httpd.conf in the typical locations for Ubuntu and for the Apache httpd project (which may differ) or just use locate and find as above.

Sometimes the main config file refers to other config files for vhosts. You can work this out by identifying the main config file.

Chronic cases

Sometimes, old servers run a variety of webserver daemons. In that case it can take a while to find them all and work out where their config files are. A combination of the techniques above should eventually succeed.

You can find what programs are listening on port 80 etc using netstat -lntp. Often, locating the binaries is a useful pointer to a directory tree that contains the config files.