For something like Mozilla, if it comes packages as a binary tarball, you can extract it in your home directory and use it. Firebird allows this, so does Thunderbird.
When compiling, you can set an option to compile the application in your home directory rather than /bin or /usr/bin or /usr/local/bin for example.
I don't know about installing RPMs without being a root user. I believe you have to be root to install RPMs. But a normal user can install applications to their home dir if they are binary tarballs or source tarballs. Just change the target directory of the build.