The JFBConnect tables are named xxx_jfbconnect_... where xxx is your Joomla DB prefix. There are 2 tables in all versions up until 4.0:
xxx_jfbconnect_user_map
xxx_jfbconnect_config
4.1.0 added 2 more tables:
xxx_jfbconnect_requests
xxx_jfbconnect_notifications
For doing the database transfer of those tables, if you have command line access, you'll want to look into "mysqldump". There are other tools though, including Akeeba, which can move specific tables from one Joomla site to another.
Finally, for running on your test domain, if it's a subdomain of the main domain you already have the app on, it should run without any problem. If it's a different domain, you should get a new app just for that domain and to test with. Once it's all working there, move to the live site and just use the live site api/secret key and everything else will just work.
Hope that all helps!
Alex