tomcat7's CATALINA_HOME in Debian/Ubuntu
It is under /usr/share/tomcat7/.
One way to find the location is to look into the service startup script /etc/init.d/tomcat7. You will find these lines in it:
NAME=tomcat7
...
CATALINA_HOME=/usr/share/$NAME
Edit:
In Ubuntu 20/Tomcat 9, it is under /usr/share/tomcat9/.
You can find the location in the systemd service file /usr/lib/systemd/system/tomcat9.service:
Environment="CATALINA_HOME=/usr/share/tomcat9"