How can I determine which paths are required?
When you login to your computer the loginwindow.app creates environment with the PATH variable set to:

/usr/bin:/bin:/usr/sbin:/sbin


This PATH is available to all applications that you start. When you start the Terminal.app this PATH is inherited by your shell which is started as a login shell. The PATH is then processed by the path_helper application to produce the PATH:

/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/bin:/usr/X11/bin


The system is designed for an application package or an administrator to add paths by creating text files with adsolute paths in the directory /private/etc/paths.d. You may find other executables located in

/private/etc
/usr/lib
/usr/libexec
/usr/local # should have the same hierarchy as /usr
/opt
/opt/local # macports package manager
/sw # fink package manager
/System
/Library


Tools may also be found in bundles and frameworks.

/Applications/LibreOffice.app/Contents/MacOS # the path to run open office headless
/System/Library/PrivateFrameworks/Apple80211.framework/Versions/A/Resources # the path to the commandline airport utility 


There are no set rules and an administrator could design his/her own prefix to install software. 

The goal of an administrator is to set the global path as small as possible then expand the path as needed according to a user's privileges, in their own startup files.