1. Can you install and configure two versions of the Oracle client running on the same (Windows) server?
2. If so, how? Note, the applications are running as Windows services and logon as the "Local System" account.
Unfortunately, upgrading the older application to the 10g Oracle client won't happen until sometime in the far off future!
Generally speaking you can have many installations on a single computer using "multiple oracle homes". During the installation of 10g you will be prompted to install into an existing home or a new one - you must define a new one for each new installation of the software and only pick an existing one when installing patches.
Unfortunately, some releases of Oracle 8 do not support multiple Oracle Homes. I don't recall which versions so check the installation guides on the Oracle 8 distribution media. You will notice the same limitation with certain really old versions of Oracle Forms and other developer tools.