Practical way to do website with multiple sections.

There are all sorts of possibilities with Apache site configurations. You can use aliases, subdirectories or whatever. Depends on what you actually *want* to achieve - which you don't say.
See the Drupal troubleshooting FAQ if you are having trouble with Drupal taking over subdirectories - there is a quick fix there.

However, the cleanest for me is subdomains.
If you own the name mysite.com you also own *.mysite.com meaning it's probably trivial to set up project1.mysite.com, project2.mysite.com using apache virtualhosts (see the apache docs - this is not a Drupal question). You may need wildcard DNS settings - but that's probably already set up. Depends. You may need help from a sysadmin.
AND you can either direct them all at the same Drupal install and run multisites (best for Drupal) or you can point them at any folder you like (best for independence)

I can't imagine any reason you'd choose Tomcat over a stock Apache+PHP distro.

Drupal doesn't use the filesystem to create URL hierarchies. Therefore there is no need to use rewrite rules to have separate site sections under different URLs. The path module in core will allow you to set any path you choose for a node. The book module in core will allow you to create a node hierarchy, as will the node_hierarchy module. Check out the pathauto module to automatically assign paths based on a node's location in the hierarchy.

The second issue won't be relevant if you do what I said above. However, if you already have two existing Drupal sites that you want to work this way, here's how I do it:
Install Drupal in some directory, say /drupal
Configure Drupal multisite for the two sections.
Create symlinks from the top level to /drupal (i.e. section1 -> drupal, section2 -> drupal)

For the third issue, I'm not really sure what you are asking. Drupal is a PHP application, Tomcat is a Java HTTP server. There is a PHP implementation written in Java, but I doubt it will perform better than mod_php or PHP in FastCGI.