I know that /usr directory is mounted as read-only in default in a sparse root zone. Is there no way to give specific users write privileges to /usr in a sparse root zone either using Solaris's role privileges or using sudo? Is it possible remove /usr from inherit-pkg-dir entries?
No. This isn't a file permission issue; it's a characteristic of the file system itself. This is similar to asking if there's a privilege to give a user the ability to write to a read-only CDROM device. Even if you could do this, you _really_ wouldn't want to. Any change to /usr in this zone would affect every other sparse root zone as well as the global zone. A root user within this zone could completely trash your machine. Not after the zone has been installed. If you were allowed to do that, you would end up with a zone that has no /usr directory at all.
But is there a way to install zone with its own /usr that is not taken from global zone right?
Yes, but that happens at initial zone install time by creating what is called a "whole root" zone. You can't change this after install, it isn't just /usr there are other directories as well the zone is fundamentally installed a different way. That's a whole root zone. http://docs.sun.com/app/docs/doc/817-1592/6mhahuoor?l=en&a=view