I am hoping to upgrade my LMS from 3.1 to 3.2 very soon. I have created a backup using NMSROOT/bin/perl backup.pl, and that has completed successfully (took a few tries, but it worked).
With this backup that I now have, can I restore it once I install 3.2 (I will be doing a clean install, not an upgrade)
Yes. And you should reboot before starting the 3.2 installation.
Yes. See https://www.cisco.com/en/US/docs/net_mgmt/ciscoworks_lan_management_solution/3.2/data_migration/guide/dmg_win.html#wp1146241 for more on performing a remote data migration.