1) When I press the button (inside the chat room) to launch the chat room into a new window I am autologged out of the gamedev site.

2) A recent thing (to my awareness) is random kick events where the user is autobooted - the user is informed by chat log that they have been kicked, while the chat log in the room itself simply shows the user has left the room. There seems also to be a time discrepancy with the notification of said event as you may experience nothing happening in chat find yourself kicked only to re-enter the room and discover that chat has been talking longer than the kick event notice.

Using Google Chrome

The chat is set to auto-kick anyone who is inactive for 10 hours - could that be the cause of problem #2, or does it seem to be more frequent?

I've definitely had it happen to me a few times when I've been gone for less than 10 hours.

Blue got kicked earlier with approximately 15-20 minutes inactivity and similarly with others over different time periods - so definitely not a case of the 10 hour window.