PATH is typically set in /etc/profile or any of the scripts that it calls. Also possibly in /etc/bashrc as well as . files in the user's home directory. Which files will vary depending on what shell you're using. See the manual page for your shell.
1. There's no such thing as Redhat 9. (Redhat ended with 8 and jumped to RHEL 3. Perhaps you meant Fedora 9?) In general you should have any "compatibility" packages installed. See ftp://ftp.thunderstone.com/pub/linux/
for libraries used by your version of Texis.
2. The install will set the right perms. Install to a test location if you want to see every perm on every file. The biggest key is that the /usr/local/morph3/texis/testdb directory and all files in it should be owned by your Texis dba account, whatever you've chosen that to be. And the files monitor, rmlocks, gw, texisd, and anytotx should be owned and setuid to the Texis dba account. vhttpd should not be setuid as it will have to run as root if you use a port under 1024. The vhttpd.conf file says what user it will run Texis as.
3. Texis should never be run as root. Otherwise it doesn't matter as long as item 2 is satisfied.
4. It doesn't care how it's accessed or by how many different names the machine it's on is accessed.