Hi Charlie,
Yes, Moodle can be installed and used in a standalone computer. Many of us have it that way for doing tests and getting familiar with new versions, or for development purposes. Exactly... about which courses do you want the reports? If the standalone computer is not connected to any network, are users then going to log in the Moodle installed in it? Anyway, putting aside clearly network-related activities, Moodle works the same in both offline and online setups.
Ok, yes, you can have three teams or groups of students, each having access only to a particular set of resources:
http://docs.moodle.org/25/en/Available_for_group_members_only
or activities:
http://docs.moodle.org/25/en/Groupings
There is a Timeout (sessiontimeout) option (Administration > Server> Session handling) that allows to extend the timeout period for up to 4 hours. I don't know if this limit can be omitted or changed; I guess it could be done by modifying the code.