I installed PHP (cgi/cli/sqlite/libcURL) with lighthttpd and also received the same issues. I fixed them with a few changes.

I uninstalled the broken(?) sqlite and libcURL modules. I'll mess with them later.

Create a directory with the proper permissions:

$mkdir /home/user/public_html
$chmod 644 /home/user/public_html

I had to edit /etc/lighttpd.conf as root:

Change this line to your newly created directory:

server.document-root = "/home/user/public_html/"

Change the CGI line to:

cgi.assign = ( ".pl" => "/usr/bin/perl", ".cgi" => "/usr/bin/perl", ".php" => "/usr/bin/php" )

Then stop/start the daemon:

#/etc/init.d/lighttpd stop
#/etc/init.d/lighttpd start

Then I uninstalled the PHP modules throwing errors (sqlite and libcURL) and deleted the .ini files inside of /etc/php5/conf.d

I then created a simple Hello World sorta PHP file in my public_html/ and it worked.