Apple - Mac OS/X won't let me mkdir /usr/share/tomcat6, even as root

That's because the /usr folder, with the exception of /usr/local, is protected by System Integrity Protection (SIP).

Check this Apple support page for more details on SIP. This answer gives details on how to disable SIP and more info.

Locally-installed software belongs in /usr/local, not /usr. This has been a good idea roughly forever, and starting in OS X El Capitan, this is enforced by System Integrity Protection. It's possible to disable SIP, but really it's better to do things right and use /usr/local/share instead.