1. I think extending the 'focus period' from a day to a week could be beneficial -- it would potentially allow more people to join in, as well as enabling conversations to develop more fully over time.

 2. I'm very much in favour of retaining two separate rooms -- one where there's a specific focus every week, and one for more general (but still on-topic) chat, including discussion of specific questions or topics that spin-off from specific questions.

My reasoning for keeping two rooms? 

 - If we're defining a weekly focussed chat topic, IMO that topic ought
   to be (pretty-well) uninterrupted by random other things -- so that
   somebody who has a burning interest in the topic doesn't have to
   negotiate the confusion that is an unfocussed chat-room. (I speak
   from experience elsewhere on SE where it can be impossible to follow
   the threads of numerous conversations going on in parallel).
   
 - We ought to have a place for spontaneous chat and news items -- it's impossible to
   predict what topics will arise on the site that excite interest but
   not of the kind that immediately generate answers.  As the site
   grows, this general chat-room will (ought) to become much busier, but
   the chats will be of shorter duration and easier to follow in
   parallel. If something develops a longer life of its own, there's an
   option to create a new chat room but those evanesce with time once
   they cease to be used, so are potentially less useful.
   
 - It will be much easier to keep them separate now than try to separate
   them in future, when the volume of chat (I hope) becomes large enough
   to warrant it.

I don't much care what the two rooms are called, but Roots sounds a bit more general to me, and I'd prefer to re-brand "Genealogy Weekly Chat" to something snappier but clearly related. ("Roots" and "Roots Focus"? "Roots" and "Roots Buzz"? Open to suggestions...)

I'd be very happy to announce each week's new topic during the ending Saturday (especially if it didn't involve me cudgelling my brain to come up with something new in the absence of community suggestions). 

ETA: If you agree with "two rooms", vote for http://meta.genealogy.stackexchange.com/a/1854/104 to keep all the votes in one place.