How do I install Tomcat on Linux without getting "permission denied" errors?
I can tell you a common solution of installing servers programs with linux:
It's common to create a user (example "tomcat") and install the software using this user.
So, the "tomcat" user will have permission to read/write all the tomcat_home directory (also logs dir, lib etc...), after installing, you'll need to use the "tomcat" user to start the service.
To create a new user, you can use "useradd" command. I use to create a directory un /usr/local/tomcat then with root account I change the owner to the new user "tomcat" (chown command). After, with root account you can switch the user (using "su tomcat" command) to install tomcat and in future you can use the user "tomcat" to run the server.