I have a question about
Private chats work in
Pidgin, all fine.
But how can I get an existing conference in
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Good question, unfortunately it's not supported by Pidgin yet. Pidgin uses the XMPP standard to talk to Google Hangouts, Google Chat, Google Talk, etc. However, Google hangouts with multiple people (conference room) does not use XMPP. Using the web interface or mobile app is probably your best option right now.
A plugin is available for pidgin for Google hangouts called: Hangouts Plugin for libpurple https://bitbucket.org/EionRobb/purple-hangouts/