What Apache modules are required? bcmath calendar ctype curl date dom ftp gd hash iconv libxml ....

i think you're better off doing some serious analysis to properly identify why your sites are slow and where your bottleneck really exists. it may not necessarily be your apache/php, it could be crawling when accessing your database. If this is the case, you may need to experiment with caching. but you'll have to spend some time identifying exactly where your bottleneck exists.

imho apache + php + mysql is not the best option if you're limited to a single server configuration. if you have been given significant resources to setup some load balancing / cluster configuration, then apache would be fine for serious drupal traffic.

i think if you're running a single server, lighttpd + fastcgi + php + mysql is way more efficient.

having said that, my php is comipled with these options:

'./configure' '--enable-fastcgi' '--enable-discard-path' '--enable-force-redirect' '--with-curl' '--with-curl-dir=/usr/local/lib' '--with-gd' '--with-gd-dir=/usr/local/lib' '--with-gettext' '--with-jpeg-dir=/usr/local/lib' '--with-mcrypt' '--with-mysql=/usr' '--with-pear' '--with-png-dir=/usr/local/lib' '--with-xml' '--with-zlib' '--with-zlib-dir=/usr/local/lib' '--with-openssl' '--enable-bcmath' '--enable-calendar' '--enable-ftp' '--enable-magic-quotes' '--enable-sockets' '--enable-track-vars' '--enable-mbstring=all'

However i think you really only need:
mbstring
curl
gd
mysql
mcrypt
jpeg
png
xml

but not completely sure. my configuration above suits me fine. not sure if if it can be of any help. If you using php as a module in apache, you can eliminate the fastcgi part.

also, you can search around these forums for ideas on how to improve drupal's performance. this is a topic that has been extensively discussed.