Falling, because technically the Czech government of last week is still the current Czech government - and will continue to be until a replacement can be found. So the current Presidency of the European Council is not so much a lame duck as a duck in a deep coma which is dependant on life support. Of course, it was never that much of a presidency - indeed it reminds me of the "Barroso ghost" picture on Anyone but Barroso - though, granted, Barroso is incorporeal through choice.
The CSSD, the main opposition party, doesn't fill me with much hope - it is unlikely to be able to form a government, so it seems that the fall of the government was due to the CSSD opposing it for opposition's sake. Their actions seem to be politically irresponsible, though I don't know much about Czech politics. They have said that they are happy to see the current ghost-government carry on until the presidency passes on to Sweden. I can't make up my mind whether this is a good thing or not.
In the end, I hold pretty much the same views expressed in this Entangled Alliances article: the presidency can be a powerful motor for change and a vehicle for effective leadership, but it largely depends on the political stability of the leader, and on personality. Which is never a good basis for good governance.
Czech president Klaus' position has been strengthened by the fall of his country's government: after all, it's now much easier to ignore/speak out against/plot against the policies of the ghost-government which he doesn't like. His influence over the passage of the Lisbon Treaty is likely to grow (though it was effectively blocked before anyway), despite polls showing that his view on the matter are out of touch with the Czech public.
(As a side note, whatever the truth of the stories of Russia's influence on Klaus (or indeed Russia's influence through Klaus), Klaus and his allies must be fought on the battleground of ideas and policy, not on "playing the man" (or woman). In any case, I'm sure that Klaus has held the views he has independently of any Russian influence, and I doubt that he needs political pointers on playing the system. If there is any Russian influence on Klaus, I'd say it is just egging him on.)
So, basically it doesn't matter much, but only because the situation was already quite bad in a way that mattered quite a lot. The point has been made that this shows that the Lisbon Treaty is needed for the effective functioning of the EU. I largely agree: it seems that over the last 6+ months the weaknesses of the EU have shown the need for reform in practically all areas, and although imperfect, the Lisbon Treaty does remedy some of the problems, to some degree.